
Class __jr§^^ 
Boole 



F55 



GapiglitlJ", 



CflEffilOHI DEPOSm 



ESSENTIALS OF 

MECHANICAL DRAFTING 

ELEMENTS, PRINCIPLES, AND METHODS 

WITH SPECIFIC APPLICATIONS IN WORKING DRAWINGS OF 
FURNITURE, MACHINE, AND SHEET METAL CONSTRUCTION 

A MANUAL FOR STUDENTS 

ARRANGED FOR REFERENCE AND STUDY IN CONNEC- 
TION WITH COURSES IN MANUAL TRAINING, INDUSTRIAL, 
HIGH, AND TECHNICAL SCHOOLS 



BY 

LUDWIG Ij-RANK 

INSTRUCTOR IN DRAWING, HIGH SCHOOL OF COMMERCE, BOSTON, MASS. 
FORMERLY ASSISTANT TO THE DIRECTOR OF MANUAL ARTS, BOSTON, MASS. 
SUPERVISING DRAWING IN PREVOCATIONAL. INDUSTRIAL, AND HIGH SCHOOLS 



1917 
MILTON BRADLEY COMPANY 

SPRINGFIELD, MASSACHUSETTS 






Copyright 1917 

BY 

MILTON BRADLEY COMPANY, 
Springfield, Mass. - 



/ 



m 23 1917 



©C1A467139 



PREFACE 

The purpose of this book is to provide the student with definite comprehensive 
text and illustrations comprising the theory and practice of mechanical drafting, 
which shall effective^ supplement and give emphasis to the work of the teacher, 
and at the same time afford complete freedom in the presentation and application 
of principles to meet different requirements, conditions, and individual needs. 

In view of this purpose, and for greater convenience of reference and connected 
study of related subject-matter, the text is presented in a progressive series of 
topically arranged articles with appropriate cross references. 

This arrangement is not intended, however, as a prescribed order of study 
to be rigidlj^ adhered to, nor is the content of the text intended to supersede 
necessary personal instruction or thoughtful study on the part of the student. 

The book is designed to be used as the teacher may determine in connection 
with his own course; to conserve the time and energy usually expended in repeti- 
tion; to secure a systematic study of such text and illustrations as relate to the 
oral presentation; and to enable the student to review any desired topic as 
individual need arises and to proceed with the minimum of dependence upon 
the teacher. 

It is beUeved that the content and arrangement will be found adaptable and 
adequate wherever mechanical drafting is taught; will assist the teacher in formu- 
lating specific courses; will stimulate the interest of the student by giving a 
greater appreciation of the utility and scope of the subject; and will prove an 
efficient aid in developing a working knowledge of the elements, principles, and 
methods of drafting as applied in general practice. 

The writer gratefully acknowledges his obligations to numerous engineers and 
draftsmen, and to the following directors and teachers of Boston, for many helpful 
suggestions: Mr. Arthur L. WiUiston, Director of Wentworth Institute; 
Mr. John C. Brodhead, Associate Director of Manual Arts; Mr. Edw. C. Emerson, 
Assistant Director of Manual Arts; Messrs. R. H. Knapp and E. H. Temple 
of the Mechanic Arts High School; Messrs. A. Roswall and E. M. Longfield of the 
Boys' Industrial School; Mr. A. L. Primeau, formerly of the South Boston Pre- 
vocational Center; and Mr. R. A. Day of the Hyde Park Co-operative Classes. 
Messrs. Temple, Primeau, and Day also gave valuable assistance in the prepara- 
tion of the drawings. 

LuDwiG Feank. 
Brookline, Januarv, 1917. 



CONTENTS 



CHAPTER I 
INTRODUCTION 

PAGE ART. 



1. Nature and Uses of Mechanical Draft- 

ing 1 

2. Geometric Terms, Definitions, etc. ... 2 



3. General Instructions for Working Out 

Problems 11 



CHAPTER II 

INSTRUMENTS, MATERIALS, AND THEIR USE. 



4. List of Equipment 13 

5. Care and Arrangement of Equipment 13 

6. Drawing Board 14 

7. T-Square 14 

8. Triangles 15 

9. Pencils and Writing Pens 16 

10. Needle-point 17 

11. Scales 17 



12. Protractor 19 

13. Compasses 19 

14. Bow Compasses 20 

15. Dividers 20 

16. Bow Dividers 20 

17. Ruling Pen 20 

18. Curve Rulers 21 

19. Erasers 22 



CHAPTER III 

PENCILING AND FINISH RENDERING 



20. Layout of the Sheet 23 

21. Constructive Stage of the Drawing. . . 23 

22. Finishing Stage of the Drawing 24 



23. Shadow Lining 25 

24. Line Shading 28 

25. Lettering 29 



CHAPTER IV 
GEOMETRIC CONSTRUCTION 



26. Geometric and Practical Methods. ... 33 

27. To bisect a straight Une or a circular arc 33 

28. To bisect an angle 33 

29. To construct an angle equal to a given 

angle 34 

30. To divide a straight line into any 

number of equal parts 34 

31. To divide a straight line into parts 

proportional to those of a given 
divided line 34 

32. To layoff the length of a given circular 

curve upon a straight line 34 

33 . To lay off the length of a given straight 

line upon an arc 34 

34. To draw a perpendicular to a line, 

from or through a given point ... 34 

35. To draw a Une at an angle of any 

given magnitude in a quadrant -with 
a given line 35 

36. To draw a parallel to a given straight 

hne: (a) at a given distance; or 
(b) through a given point 36 

37. To construct a triangle: having given 

(a) the sides; (b) a side and angles 
adjacent to it; or (c) a side, an adja- 
cent angle, and angle opposite the side 36 



38. To construct an isosceles triangle when 

the base and vertex angle are given 36 

39. To construct an equilateral triangle 

when the altitude is given 37 

40. To circumscribe a circle about a given 

triangle 37 

41. To inscribe a circle within a given 

triangle 37 

42. To inscribe an equilateral triangle 

within a circle 37 

43. To circumscribe an equilateral triangle 

aboutacircle 37 

44. To construct a parallelogram when 

two sides and the included angle 
are given 37 

45. To construct a square on a given side 37 

46. To inscribe a square within a circle. . . 38 

47. To construct a square on a given 

diagonal 38 

48. To circumscribe a square about a circle 38 

49. To inscribe a regular pentagon within 

a circle 38 

50. To construct a regular hexagon on a 

given side 38 

51. To inscribe a regular hexagon within 

a circle 38 



52, 
53, 
54. 
55. 



56. 
57. 



58. 

59. 
60. 



PAGE 

To construct a regular hexagon on a 

given long diagonal 38 

To inscribe a regular octagon within a 

circle ■ 39 

To circumscribe a regular octagon 

about a circle '. 39 

To construct any regular polygon. 

General methods: having given (a) a 

side; (b) the circumscribing circle; 

or (c) the inscribed circle 39 

To construct a polygon similar to a 

given polygon, upon a given side. . . 40 
To plot a figure similar to a given 

figure by means of a base line or 

center line, and offsets 40 

To draw a tangent to a circle through 

a given point 41 

To draw a tangent to two given circles 41 
To draw a circular curve of given 



ART. PAGE 

radius tangent to a given circular 
curve and to a given straight line, 
or to two given circular curves 42 

61. To draw a circular curve tangent to 
two given straight lines: having 
given (a) a point of tangency; or 
(b) the radius 42 

62' To draw a circular curve tangent to 

three straight lines 42 

63. To draw circular curves tangent to 

two given parallel straight lines and 

to each other, at a given point 43 

64. To draw an ellipse when the axes are 

given: 

(a) By Focal Radii 43" 

(b) By Trammel Method 43 

(c) By Revolution of a Circle 43 

(d) By Parallelogram Method 43 

(e) By Circular Arcs 44 



CHAPTER V 
ORTHOGRAPHIC PROJECTION 



65. General Principles 45 

66. To draw the front, top, and side views 

of a rectangular object 49 

67. Objects Having Surfaces Obhque to 

the Co-ordinate Planes 51 

68. Objects Having Curved Surfaces .... 51 

69. Projections upon Oblique Auxiliary 

Planes 52 

70. Revolution of Surfaces 54 



71. True Length and Position of Lines 

Oblique to the Co-ordinate Planes . . 55 

72. Objects Obhque to the Co-ordinate 

Planes 56 

73. Partial Views, and Use of Auxihary 

Views in Determining Required 
Views 58 

74. Rules Governing the Position of Lines 

and Surfaces Relative to Any Two 
Perpendicular Planes of Projection. 61 



CHAPTER VI 
PLANE SECTIONS 



75. Principles and Methods 62 

76. Objects Having Plane Surfaces 62 



77. Objects Having Curved Surfaces . . 



64 



CHAPTER VII 
INTERSECTION OF SURFACES 



78. Principles and Methods 66 

79. Objects Having Plane Surfaces 67 



80. Objects Having Curved Surfaces . 



CHAPTER VIII 
DEVELOPMENT OF SURFACES 



81. Principles and Methods 71 

82. Objects Having Plane Surfaces 72 

83. Objects Having Cylindric or Conic 

Surfaces 73 



84. Objects Having Double Curved or 

Warped Surfaces 77 

85. Intersecting Surf aces 78 



CHAPTER IX 

MECHANICAL PICTORIAL DRAWINGS 



86. Character and Purpose of the Drawing 80 

87. Isometric Projection 81 

88. To draw the isometric of a rectangular 

object 81 



89. To draw the isometric of an object 

involving non-isometric figures 83 

90. Oblique Projection 86 

91. Shade Lines, Shadow Lines, and Line 

Shading 87 



CHAPTER X 

WORKING DRAWINGS 



92. CharacterandPurposeof the Drawing 89 

93. Tjfpesof Drawings 89 

94. Position of Object and Arrangement 

of Views 92 

95. Selection and Number of Views, etc. . . 92 

96. Center Lines 94 

97. Conventional Representations 94 

98. Sectional Views 96 

99. Broken Views 98 



ART. PAGE 

Standard Sizes of Sheets and Scale of 

Drawings 99 

101. Dimensioning 100 

102. Lettering 105 

103. Shadow Lining and Line Shading. . . 108 

Sketching 108 

Making Scale Drawings 109 

Tracing and Blue-printing 115 

Checking Drawings 120 



100. 



104. 
105. 
106. 
107. 



108. Reading Drawings 120 



CHAPTER XI 
HELICAL CURVES, THREADED PARTS, AND SPRINGS 



109. HeUces 122 

110. Screw Threads 123 

111. Pipe Threads 128 



112. Bolts 128 

113. Screws 131 

114. Springs 132 



SYMBOLS AND GENERAL ABBREVIATIONS 



II parallel; ||s parallels 

± perpendicular; J_s perpendiculars 

Z. angle; Zs angles 

A triangle; As triangles 

G circle; Os circles 

C. L. center line; C. Ls. center lines 



diam. diameter; diams. diameters 

hor. horizontal; hors. horizontals 

pt. point; pts. points 

rad. radius 

St. straight 

vert. vertical; verts, verticals 



ESSENTIALS OF 
MECHANICAL DRAFTING 



CHAPTER I 
INTRODUCTION 

1. Nature and Uses of Mechanical Drafting. Mechanical drafting or 
mechanical drawing is the art of making the conventional representations used by 
engineers, architects, and inventors in working out and recording the details of 
their constructive designs, and the means by which ideas of the exact form or 
shape, dimension, and arrangement of parts in objects of a structural charac- 
ter are universally expressed and made intelligible to others. 

Mechanical drafting enables constructive work of any kind to be carried on 
with accuracy and economy of time and material, and takes the place of lengthy 
verbal description which would fail to express with clearness and exactness the 
definite information required by the workman. 

It will be seen from Fig. 179 that certain general peculiarities of the form and 
structure of an object may be understood from an ordinary pictorial representa- 
tion, but that it cannot show the exact form, size, and relation of all the lines 
and surfaces; hence the necessity for mechanical drawings which show all hidden 
as well as visible parts of an object as they are and not as they would appear 
to the eye. 

Mechanical drafting is thus the graphic language of the constructive or 
mechanic arts, and ability to read or comprehend mechanical drawings is of as 
great importance to the workman, builder, and manufacturer as ability to make 
such representations is to the designer or draftsman; and a knowledge of general 
drafting principles is of value to almost all men irrespective of their vocations. 

Because of the exact nature of the facts which it is intended to record or convey 
the drawing is generallj^ executed with the aid of instruments. 

The mechanical character of the representation, together with its purpose 
and the usual means of execution, gives mechanical drafting its name. 

Machine drafting, architectural drafting, and engineering drafting are specific 
applications of mechanical drafting. 

A mechanical drawing properly dimensioned in figures and prepared as a 
guide in constructing the object is called a working drawing. 



2 ESSENTIALS OF MECHANICAL DRAFTING 

2. Geometric Terms, Definitions, etc. Geometry is the science which 
describes the definite figures (forms or shapes) upon which all objects however 
complex are based, and the principles and methods by which these figures may be 
measured and graphically constructed. 

Geometry is thus fundamental in mechanical drafting and in all the con- 
structive arts. 

The terms defined in this chapter are commonly involved in both. 



(a) General Definitions. A physical solid or material object occupies a certain portion of 
space and has shape, size, weight, color, etc. Geometry is concerned simply with the shape and 
size of the space which a physical solid occupies or is conceived to occupy; hence — a geometric solid 
is a limited portion of space. 

A soKd has dimensions or extent in three principal directions at right Zs to each other; namely, 
length, breadth (or width), and thickness {height, altitude, or depth). 

The boundaries of a solid are called surfaces. 

A surface has only two dimensions: length and breadth (or width). 

The boundaries of a surface are called lines. 

A line has only one dimension : length. 

The hmits or ends of a line are called points. 

A point has position but no dimension. 

Points, lines, and surfaces may be considered as apart from a solid, or as combined in any con- 
ceivable figure; also a line may be imagined as generated by a point, a surface by a line, and a solid 
by a surface, in motion. 

Similar figures are those having the same shape; equivalent figures those having the same size; 
and equal or congruent figures, those having the same shape and size. 

A figure that Ues wholly in one plane is a plane figure. (See Art. (h).) A figure whose lines are 
straight is rectilinear; one whose lines are curved is curvilinear. 

The axis of a figure is a st. line which passes through its center, and about which it is symmetrical 
or balanced. 

An axis of revolution is a st. line about which a figure is revolved. 

When two lines, two surfaces, or a line and a surface meet or cross they are said to intersect or 
cut each other and the pt. or line in which they intersect is their inter'section. 

A bisector is a pt., line, or plane which divides a figure into 

Fig. I two equal parts, that is, bisects it. To trisect is to divide into 

three equal parts; to quadrisect, into four equal parts. 



(b) Lines. A straight or right line has the same direction 
throughout. Fig. 1. 

A curved line or curve is one no part of which is straight. 
Fig. 2. 

A reversed curve is one whose direction of curvature changes. 
Fig. 3. 

A horizontal line is one that is level throughout. In draw- 
ing, the term is appUed to a st. line drawn from left to right, 
without slant. Fig. 1. 

A vertical line is one that is upright or plumb. In drawing, 
the term is applied to a st. fine drawn from bottom to top, with- 
out slant. Fig. 4. 

An oblique line is one that slants. Fig. 5. 

Two lines having the same relative direction are parallel to 
each other. They are the same distance apart throughout. 
Fig. 6. 




Fig 4 



Fig. 




Fig. 6 



INTRODUCTION 



Two St. lines which extend from the same pt. or which would intersect if extended, are said to be 
at an angle with each other (Fig. 7), and are perpendicular, or oblique, to each other according as 
the included Z is a right Z or an oblique Z. See Art. (c). 

Two curves, or a st. line and a curve, are tangent to each other when they touch in but one pt. 
and cannot intersect. The pt. is the point of tangency. Figs. 11(a), 12, 14. 

An ordinate or offset is the J_ distance of a pt. from a given st. hne, or plane, of reference. A-1, 
B-2, etc., Fig. 103. 

Co-ordinates are ordinates of the same pt., measured || to two, or three, mutually J. hnes, or 
planes, of reference. C-Cv, C-Ch, C-Qp, Fig. 118. 



\ 





Fig. 7 



V. Z 



1 



[Fig. 8 ^~- Fig. 9 



(c) Angles. These definitions refer to plane Zs only. 

An angle is the opening between two st. lines, called the sides of the angle, which extend from 
a pt. called the vertex. BAG, Fig. 7. An angle may be considered as generated by a st. Une revolved 
about one of its ends. 

The size of an angle depends upon the relative direction of the sides and not upon their length. 
When the sides extend in opposite directions, so as to lie in the same st. line, the Z is a straight angle. 
When the directions are such that the Zs formed by extending the sides beyond the vertex are 
equal, each Z is a right angle. A right Z is equal to half a st. Z. Fig. 8. 

An Z less than a right Z is an acute angle (Fig. 7) ; one greater than a right Z and less than a 
St. Z is an obtuse angle (Fig. 9); one greater than a st. Z and less than two st. Zs is a reflex angle. 
Note that two st. lines extending from a pt. always form two Zs, as Za and Zb, Fig. 9. 

Angles other than right and straight Zs a,ie oblique angles. 

Two Zs having the same vertex and a common side between them are adjacent angles. DCF 
and FCE, Fig. 10. 

When two st. lines intersect, the opposite or non-adjacent Zs are equal and are called vertical 
angles. GCD and FCE, also Zs DCF and ECG, Fig. 10. 




An angle is said to be measured by an arc, described from its vertex as center and included by its 
sides. The unit of measure is an angle whose arc is a degree (5-5^ of a circumference). (See Art. (d).) 
Thus a St. Z is one of 180°, a right Z one of 90°, and the whole angular space about a pt. in a plane 
equals 360°. See Fig. 10. 

Lines can be drawn in four directions from a given pt., at the same given Z with a given line; 
thus in Fig. 10, C-D, C-E, C-F, and C-G each make an Z of 30° with A-A. Each of these lines makes 
two Zs with the hor. A-A and two with the vert. B-B. Thus C-D makes Zs of 150° and 30° with 
A-A, and 60° and 120° with B-B. 



4 ESSENTIALS OF MECHANICAL DRAFTING 

In speaking of the Zs formed by two lines, the lesser is the one usually named, and, unless other- 
wise stated, a line at 15°, 30°, 45°, etc., is understood to mean 15°, 30°, 45°, etc., with the hor. direction. 

(d) Curvilinear Figures. A circle* is a plane figure bounded by a curve called its circumference, 
all pts. of which are equidistant from a pt. within called the center. Fig. 11(a). 

Any part of a circumference is an arc. 

A St. line intersecting a circular curve in two pts. is a secant, F-G. A st. line joining two pts. 
in the curve is a chord, H-I, A-B. 

A chord through the center is a diameter. 

A straight Une from the center to the curve is a radius, C-A, C-D, C-E. 

A segment is a portion of a O boundetkby an arc and its'chord. 

A segment equal to half a O is a semicircle*. 

A sector is a portion of a O bounded by an arc and two radii. 

A sector equal to one-fourth of a O is a quadrant*. 

An Z formed by two chords from the same pt. is an inscribed angle. Fig. 11(b). An angle is 
inscribed in a segment when its sides join the ends of the arc. 

The Z included by two radii is a central angle. 





Fig. 13 



Fig. 12 



N-^-^b) 




c J 



Fig. 14 



The circumference of a O is conceived to contain 360 equal parts called degrees (360°), each degree 
60 equal parts called minutes (60'), and each minute 60 equal parts called seconds (60"). 

A St. tangent is X to a rad. at the pt. of tangency. J-K, Fig. 11(a). The pt. of tangency of two 
circular curves is in their line of centers. Fig. 12. 

Two circles or arcs having the same center are concentric. Fig. 13(a). 

Two circles not having the same center are eccentric when one is within the other. Fig. 13(b). 

An ellipse* is a plane figure bounded by a curve called its circumference, the sum of the distances 
of every pt. of which, from two pts. within, called the focuses or foci, is constant. Fig. 14. 

A pt. midway between the foci is called the center. 

A St. line joining any two pts. in the curve is a chord, 

A chord through the center is a diameter. 

A diam. containing the foci and center is the long or major axis. A diam. ± to the major axis 
is the short or minor axis. The major and minor axes are the longest and shortest diameters of the 
eUipse and bisect each other at the center. , 

A St. line from either focus to any pt. in the curve is a focal radius. 

The sum of the focal radii of any pt. is equal to the major axis. 

A st. tangent bisects the Z between one focal rad. and the other extended at the pt. of tangency. 
When one diam. is || to the tangents at the ends of another, the diams. are conjugate to each other. 



*The terms " circle," " semicircle," " quadrant, " and " ellipse " are also used to denote merely the curve. 



INTRODUCTION 



In general any side may be 



(e) Polygons. These definitions refer to plane polygons only. 

A polygon is a plane figure bounded by st. lines called its sides. The sum of the sides is the 
■perimeler; the Z s formed by the sides are the angles; and the vertices of the / s, the vertices of the 
polygon. Figs. 15-30. 

A polygon is equilateral when all of its sides are equal (Figs. 15, 24, 27) ; equiangular when all its 
Zs are equal (Fig. 21); regular when both equilateral and equiangular (Figs. 15,22, 28-30) ; otherwise, 
it is irregular. 

The hase of a polygon is the side upon which it is supposed to rest, 
considered as the base. 

The altitude is the ± distance between the base, or base extended, and the farthermost vertex 
or side. A-B in Figs. 15, 17, 19, 23, 26. 

A diagonal is a st. line joining any two non-consecutive 
vertices. A-B, Fig. 21; A-B and A-C, Fig. 29. 

A polygon is inscribed in a O when all its sides are chords 
of the O, and circumscribed about a O when all its sides are 
tangents of the O. Also the O is circumscribed about the 
inscribed polygon and inscribed in the circumscribed polygon. 
Fig. 28. 

The center of a regular polygon is the center of the inscribed 
or circumscribed O . 

The rad. of the inscribed O is the apothem, and the rad. of 
the circimiscribed O the radius of the potygon. Fig. 28. 

In a regular polygon of an even number of sides the diameter 
of the inscribed O is often called the short diameter, and that of 
the circumscribed O the long diameter of the polygon. 

The sum of the Zs of any polygon is equal to two right 
Zs (180°), taken as manj^ times less two as the figure has sides. 
See Fig. 20. 

The Z included by the radii to the ends of any side of 
a regular polygon is called the angle at the center. It is equal to 
360° divided by the number of sides. Fig. 30. 

A polygon of three sides is a triangle; of four, a quadrilateral; 
of 'five, a pentagon; of six, a hexagon; of seven, a heptagon; of 
eight, an octagon; of nine, a nonagon; often, a decagon. 

(f) Triangles. Triangles are classified according to relative 
length of sides; as equilateral, isosceles, and scalene. 

An equilateral triangle has aU sides equal; it is also equi- 
angular. Fig. 15. 

An isosceles triangle has two sides equal; two Zs are also 
equal. Fig. 16. The equal sides are usually called the sides, and the other side, the base. 

A scalene triangle has no two sides equal; its Zs are also imequal. Fig. 17. 

Triangles are classified according to kind of Zs; as right, obtuse, and acute. 

A A is a right triangle when one Z is a right Z. Fig. 18. The side opposite the right Z is 
called the hypotenuse, the others are usually called the sides. 

A A is an obtuse triangle when one Z is obtuse. Fig. 19. 

A A is an acute triangle when all Zs are acute. Fig. 20(a). 

The Z opposite the base of a A is called the vertex angle, and its vertex , the vertex of the triangle. 




Fig. 20 



(g) Quadrilaterals. A parallelogram is a quadrilateral whose opposite sides are ||. Figs. 
21-24. 

A rectangle is a right-angled parallelogram. Figs. 21, 22. 

A square is an equilateral rectangle. Fig. 22. 

A rectangle whose opposite sides only are equal is often called an oblong. Fig. 21. 

A rhomboid is an oblique-angled parallelogram. Figs. 23, 24. 



ESSENTIALS OF MECHANICAL DRAFTING 




Fig. 21 



Fig. 25 




Fig 30 



A rhombus is an equilateral rhomboid. Fig. 24. 

The side 1 1 to the base of a parallelogram is called the upper base, the other is the lower base. 
A trapezoid is a quadrilateral which has two sides only ||. Fig. 25. The parallel sides are the 
upper and lower bases. 

A trapezium is a quadrilateral which has no two sides 1 1 . Fig. 26. 

(h) Surfaces. A plane surface or plane is a surface such that a st. line through any two pts. in 
it lies whoUy in the surface. 

A curved surface is one no part of which is plane. If a semicircular arc be revolved about its 
chord, it will generate a spheric surface. Fig. 46. 

A St. line which moves 1 1 to its first position and constantly touches a fixed curve not in the plane 
of the line, will generate a cylindric surface. Fig. 39. 

A moving st. line which constantly intersects a fixed curve and passes through a fixed pt. not in 
the plane of the curve will generate a conic surface. The fixed pt. is its vertex. Fig. 43. 

Curved surfaces generated by st. lines are single curved surfaces. The generating line in any of 
its positions is called an element. 

A warped surface is one of single curvature in which no two consecutive elements are 1 1 or inter- 
secting. A in Fig. 158. 

A double curved surface is one generated by a curve of which no two consecutive positions are 
II; as the surface of a sphere, elHpsoid, etc. Figs. 46, 47. 

A curved surface generated by the revolution of a line about an axis is called a surface of revolution; 
as a right circular cylindric or conic surface, etc. Figs. 39, 43, 46, 47. Each pt. of the generating 
line describes a O whose plane is _L to the axis. 

A plane surface is horizontal when it is level throughout; vertical when J. to a hor. plane; oblique 
when neither hor. nor vert. 

Two surfaces, or a line and a surface, are parallel when they are the same distance apart throughout. 

Two plane surfaces which extend from the same st. line or which would intersect if extended, are 
said to be at an angle with each other; and are perpendicular, or obliqiie, to each other according as 
the included Z is a right or an oblique dihedral. 

A dihedral or dihedral angle is the opening between two planes called the /aces of the angle, which 
intersect in a st. line called its edge. 

A dihedral is measured by the plane Z formed by two st. lines, one in each face and _L to the 
edge at the same pt. 



INTRODUCTION 



A dihedral is right, acule, or ohluse according as its measure is a right, acute, or obtuse plane Z . 

A St. line and a plane are ± to each other when the line is X to every st. line through its foot 
or pt. of intersection with the plane. 

The Z a line makes with a plane is the Z which it makes with a line in the plane passing 
through its foot and the foot of a ± from any other pt. in the line. 

A plane and a curved surface, or two curved surfaces, are tangent when they touch in but one pt. 
or in one line, and cannot intersect. The pt. or line is the point or line of tangency. 

A St. line or curve and a curved surface, or a curve and plane, are tangent when they touch in 
but one pt. and cannot intersect. 

(i) Solids. The base is the plane surface of the solid upon which it is supposed to rest. 

The altitude is the ± distance between the plane of the base and the farthermost vertex or part. 

A plinth is a prism, or c\'linder, whose altitude is its least dimension. Figs. 33, 42. 

A plane section is the figure formed by the intersection of a soUd with a plane passing through it. 

A solid of revolution is one which may be generated by a plane surface revolving about an axis. 

A solid bounded by plane surfaces is called a polyhedron. Figs. 31-38. The bounding surfaces 
are its faces; the intersection of its faces, the edges; and the vertices of the faces, the vertices of the 
polyhedron. 

A diagonal of a polyhedron is a st. line joining any two vertices not in the same face, as A-B, Fig. 
31(a). 




^KD^ 





Fig. 31 



Fig. 32 



Fig. 33 



Fig 34 



Fig. 35 



(j) Prisms. A prism is a pol}'hedron bounded by two equal polygons called its bases, and by 
three or more parallelograms called its lateral faces. The intersections of its lateral faces are its 
lateral edges; the others are its base edges. Figs. 31, 32, 33, 35. 

Prisms are named from their bases; as triangular, square, etc. 

A prism whose base centers lie in a JL to its bases is a right prism. Figs. 31-33. All others are 
oblique. Fig. 35. 

A regular prism is a right prism whose bases are regular polygons. Its lateral faces are equal 
rectangles. 

A cube is a regular prism whose six faces are equal squares. Fig. 32. 

A right section of a prism is a section ± to its lateral edges. 

A truncated prism is the portion of a prism included between a base and a section oblique to the 
base. Fig. 34. 

(k) Pyhajiids. a pyramid is a polyhedron bounded by a pol}'gon called its base, and three or 
more As called its lateral faces, meeting in a common pt. called the vertex of the pyramid. 

The intersection of its lateral faces are its lateral edges; the others are its base edges. Figs. 36, 37. 

Pyramids are named from their bases; as triangular, square, etc. 

A pjTamid whose verte.x lies in a ± to the center of its base is a right pyramid. Fig. 36. All 
others are oblique. Fig. 37. 

A regular pyramid is a right pyramid whose base is a regular polygon. Its lateral faces are 
equal As. 

The altitude of a lateral face of a regular pyramid is the slant height of the pyramid. C-B, Fig. 36. 

A truncated pyramid is the portion of a pyramid included between the base and any plane section. 
Figs. 38 (a) and (b ) . When the section is 1 1 to the base, the included portion is a frust um of a pyramid. 
Fig. 38(a). 



8 



ESSENTIALS OF MECHANICAL DRAFTING 



(1) Cylinders. A cylinder is a solid bounded by a closed cylindric surface called the lateral 
surface, and two || plane surfaces called its bases. Figs. 39, 40, 42. 

A cylinder is named from its bases; as circular, elliptic, etc. The terms "right," "oblique," and 
"truncated" apply to a cylinder as to a prism. 




f 


B 


N 










D 






_^ 


— 


""^ 


^^ 


'a 












^ 


^ ^ 


c 








Fig. 39 Fig. 40 



Fig. 3e> 




Fig. 4 



Fig. 42 



A rij/ii circular cylinder may be generated by the revolution of a rectangle about one of its sides. 
Fig. 39. 

A right section of a cyUnder is a section J_ to its elements. 

(m) Cones. A cone is a solid bounded by a closed conic surface called the lateral surface, and 
a plane surface called its base. Figs. 43, 44. 

A cone is named from its base; as circular, elliptic, etc' The terms "right," "oblique," "truncated," 
and "frustum" apply to a cone as to a pyramid. 

A right circular cone may be generated by the revolution of a right A about one of its sides. Fig. 
43. The length of an element of a right circular cone is the slant height. 

(n) A Sphere is a soUd bounded by a closed spheric surface every point of which is eqmdistant 
from a pt. within called the center. Fig. 46. A st. hne from the. center to the surface is a radius; 
a St. line through the center and terminated at each end by the surface is a diameter. 




Fig. 47 



A sphere may be generated by the revolution of a semicircle about its diam. 

A plane section through the center is a great circle of the sphere. Any others are small circles of 
the sphere. 

Any great O divides a sphere into two equal parts called hemispheres. 

A spheroid (ellipsoid) is a solid which may be generated by the revolution of an ellipse about 
either its long or short diam. Fig. 47. 



INTRODUCTION 9 

(o) Areas axb Voluiies. The number of times a geometric magnitude contains a given vmit 
of measure of the same kind is the numeric measure of the magnitude. 

The ratio of two magnitudes is the quotient of their numeric measures, expressed in terms of the 
same unit. Thus the ratio of 2" to 3" is f, or 2 : 3. 

The expression of the equahty of two ratios is called a proportion. As |=|. Read 2 is to 3 
as 4 is to 6. The quantities compared are said to be in proportion or proportional, and are called 
the terms. The first and last terms are the extremes and the middle terms, the means. In any 
l^roportion the product of the extremes equals the product of the means; hence, if three terms are 
given, the fourth may be found. 

The area of a surface is its measure expressed in some iinit of surface, as a square inch, square 
foot, etc. 

The area (A) of a parallelogram is equal to the product of its base (b) and altitude (a) . A = ba. 

The area of a square is equal to the square of one of its sides (s). A = s^. 

The length of the side is equal to the square root of the area. s= V A. 

The length of the diagonal (d) is equal to the square root of 2 times the square of the side. 
d = r'2Xs- or d = sl 2l (,v¥=1.414). 

The area (A) of a A is equal to k the product of its base (b) and altitude (a). A= J ba. 

The altitude (a) of an equilateral A is equal to k the product of the side (s) and the square root 

of 3. a = ^. (V''3"= 1.732). 

The hjfpotenuse (h) of a right A is equal to the square root of the sum of the squares of the other 
two sides (b) and (c). h = T b" + c-. 

The area of a trapezoid is equal to J the product of its altitude (a) and the sum of its bases (b) 
and (b'). A=ia(b+b'). 

Any polygon may be divided into As. The area of the polygon is equal to the sum of the areas 
of its As. 

The length of the circumference (c) of a O is equal (very nearly) to 3.1416 times the lengtli of 
the diameter (d). 3.1416 is designated by the Greek letter (pi), c = Trd or 27rr. 

The area of a O is equal to tv times the square of its rad. (r). A = jit-. 

The volume of a solid is its measiu'e expressed in some unit of volume, as a cubic inch, cubic foot, 
etc. 

The volume (\) of a cube is equal to the cube of its edge (s) or third power of its dimension. 

V = s». 

The volume of a prism, or cvhnder, is equal to the product of its base (b) and altitude (a). 

V = ba. 

The lateral area (1) of a prism, or cylinder, is equal to the product of a lateral edge or element 
(e), and the perimeter (p) or circumference (c), of a right section. 1 = ep, or 1 = ec. 

The volume of a pjTamid, or cone, is equal to | the product of its base and altitude. V = Jba. 

The lateral area of a regular pjTamid, or right circular cone, is equal to J the product of its slant 
height (s) and the perimeter, or circumference, of its base. 1 = |sp, or 1 = Jsc = iiTs. 

The area of the surface (s) of a sphere is equal to the product of the circumference of a great 
O and its diam. (d), that is, 27rrd, and is equivalent to the area of 4 great Qs. s = 45n-2. 

The volume of a sphere is equal to the product of ^ of its rad. and the area of its surface, that is, 
ir X 4:^2 = J7rr3. V = i^n-s or -J7rd3. 



10 



ESSENTIALS OF MECHANICAL DRAFTING 



Decimal Equivalents of Fractions of an Inch 





W 


irV 


1 


3 


Its 


A 


A 


1 


iV 


8 


A 


A 


3 


1 1 

6T 


16 


7 


1 3 


1 


M 


4 




1 7 


fi 


H 


le 


21 
"6T 


3 


li 


i 


if 


25 


7, 


If 


lb 


1 5 


li 


1 


31 

¥T 


i 







015625 

.03125 

.046875 

.0625 

.078125 

.09375 

.109375 

.125 

.140625 

.15625 

.171875 

.1875 

.203125 

.21875 

.234375 

.25 

.265625 

.28125 

.296875 

.3125 

.328125 

.34375 

.359375 

.375 

.390625 

.40625 

.421875 

.4375 

.453125 

.46875 

.484375 

.5 



T¥ 



1 3 
1 6 



i_5 
1 S" 



1 7 
T2 


3 3 


35 

¥¥ 


M 


37 


If 


li 


H 


If 


If 


4.5 


H 


25 


49 
¥¥ 


li 


li 


53 


II 


If 


II 
59 


li 


li 


II 







.515625 

.53125 

.546875 

.5625 

.578125 

.59375 

,609375 

.625 

.640625 

.65625 

•671875 

.6875 

.703125 

.71875 

.734375 

.75 

.765625 

.78125 

.796875 

.8125 

.828125 

.84375 

.859375 

.875 

.890625 

.90625 

.921875 

.9375 

.953125 

.96875 

.984375 



INTRODUCTION 11 

3. General Instructions for Working Out Problems. In the solution of 
graphic problems clear mental images of the forms to be represented, definite 
ideas of the purpose of the drawings, and the orderly application of appropriate 
pi'inciples and working methods, are fundamental. 

Habits of accuracy, thoroughness, and neatness should be cultivated from the 
outset as the essentials of good workmanship. The value of the work lies not 
in the completed drawings, but in the knowledge and ability acquired by the 
student through his own efforts in solving problems and in striving to attain 
mastery of the principles and methods which will enable him to represent any 
form whether real or imaginary. 

Upon the presentation of a problem the student should first form a 
definite idea of what is required, the conditions and principles involved, and the 
method of construction to be employed. He should then start the drawing, 
beginning with the parts that are knoivn or given. These will suggest other 
parts. It is not necessary to imagine the complete solution before beginning 
to draw. 

Test and correct each stage of the solution before proceeding with the next. 
Upon the completion the student should make a brief notebook summary for 
future reference. 

For use of instruments and materials, see Chap. II. 

For general instructions in penciling and finish rendering, see Chap. III. 

(a) Geometric Construction Sheets. Chap. IV. Unless otherwise 
directed solve problems by practical methods whenever such are known or given. 
Within reasonable limits prove or test the constructioiis by geometric methods, 
using compasses and one triangle only; and retain all working lines. Make 
constructions as large as practicable to insure greater accuracy, and extend the 
working lines beyond the pts. as shown in the figures. 

To avoid impairing the accuracy of the constructions, it is recommended that 
they be left in pencil. 

(b) Orthographic Projection Sheets. Chaps. V-VIII. Locate first the traces 
of the planes, or equivalent lines of reference (base or C. Ls.). Then proceed 
to determine the views, beginning in general with that view or part about which 
most is known; in the case of an object, for example, the view which will show 
the largest number of the lines and surfaces of the object in their exact form and 
dimension. Instead of completing the views separately it is usually desirable 
to carry along the views of corresponding parts at the same time. Unless other- 
wise directed use practical rather than geometric methods for the constructions, 
and obtain the solutions by the aid of dimensioned freehand sketches from objects. 
See Art. 104. 

' To determine the positions of the lines and surfaces more readily, number or 
letter each pt. lightly as located, marking the corresponding views to indicate 
that they represent the same pt. of the object. Iil locating positions of centers 
or other important pts., small freehand Os may be penciled about them. When 
the drawing has been approved, these notation marks and Os should be erased. 



12 ESSENTIALS OF MECHANICAL DRAFTING 

(c) Isometric and Oblique Projection Sheets. Chap. IX. Locate first 
the axes, or reference lines, then locate the main lines or surfaces of the object. 
Gradually work from these to the more important details, then proceed to the 
smaller details. 

Obtain the dimensions from objects; exact orthographic views; working 
drawings; or from dimensioned sketches in orthographic, isometric, or oblique 
projection, as may be directed. 

(d) Working Drawings. Chap. X. Locate first the main C.Ls. of the 
views according to the layout sketch (Art. 105 (a)), then locate the main lines or 
surfaces of the object, as indicated in Fig. 209 (a), beginning with the view about 
which most is known, as in Art. (b). 

Proceed in like manner with the more important modifications or details 
of the main body (Fig. 209 (b)) and from these work down to the smaller details. 
Next, add the dimension and extension lines, arrowheads, figures, and lettering. 
Fig. 209 (c), (d). Finally indicate the section lines. Do not try to complete 
the views separately. Within practical limits those of corresponding parts 
should be carried along at the same time. Use practical methods for the con- 
structions whenever adequate. , 

Obtain the working data from dimensioned sketches made from objects, or 
as directed. 

For methods of representing the commoner forms of bolts, screws, etc., see 
Chap. XL For tables of sizes, and the construction and proportions of other 
machine details, see manufacturers' catalogs, books on machine design, and 
engineers' handbooks. For sizes, etc., of details of wood construction, see books 
on joinery, etc. 



CHAPTER II 
INSTRUMENTS, MATERIALS, AND THEIR USE 

4. List of Equipment. The instruments and materials ordinarily needed 
are enumerated in the following. A good equipment is indispensable to good 
work. If not furnished by the school, its selection should be intrusted to an 
experienced draftsman. 

Set of Instruments, consisting of Compasses 5|" (joint in both legs preferred), with Lead Holder, 
Pen, and Lengthening Bar attachments; Ruling Pen, medium; Dividers, 5"; Bow Pencil; Bow 
Pen; and Bow Dividers. 

Leads, grade 4H, for compasses and bow pencil. 

Drawing Board, IS" -x 24", is suitable for most work. 

T-square, fixed head, blade sUghtlj' longer than board. 

Triangles, one 45°, 7", and one 30° x 60°, 9" (celluloid preferred). 

Scale, 12", flat, both edges divided into full size inches, halves, 4ths, 8ths, and 16ths, and the 
first inch, or first and last, into 32ds. 

Or, a 12" architect's triangular scale: one edge divided into full size inches and fractions, to 
16ths, and the others to scales of 3", If", 1", f", i", f", J", ^", I", and 35" to 1 foot. 

(The flat scale is recommended if scales of full, half, quarter, eighth, and sixteenth sizes only 
are to be used.) 

Curve Rulers, one or two, similar to those shown in Figs. 66, 67 (celluloid preferred). 

Drawing Pencils, one hard, grade 4H, 5H, or 6H, and one medium hard, grade H or 2H. 

Erasers, one soft rubber for pencil erasing and one hard rubber for ink erasing. 

Needle-point, a fine needle inserted in a wooden handle about 3|" long, for use in fixing pts. 

Tacks, 1 oz. copper, or small thumb-tacks, for fastening paper to board. 

Tack-driver. A small screw-driver ground to a thin edge and sKghth' bent at the end will be 
suitable. 

Knife, for pencil sharpening (one with broad blade preferred). 

Lead-pointer, a strip of No. J sandpaper or No. 120 emery cloth, about f " x 4", glued upon 
a fiat strip of wood. 

Penholder and Writing Pens, for lettering, etc. (Tapering handle with cork grip preferred.) 
Pens should be medium and coarse. 

Penwiper, a piece of cloth or wash leather. 

Drawing Ink, one bottle waterproof black; wTiting ink is unsuitable. 

Drawing Paper, hard and tough, with a surface not easily roughened by erasures. For sketches 
a softer paper is desirable. Avoid rolling the paper. Sizes in common use are 18" x 24", 12" x 18" 
and 9" x 12"; also 1.5" x 22", and 11" x 15". 

PortfoUo or Binding Cover, to hold paper and drawings. 

Notebook Cover, with loose sheets, about 6" x 9". 

Cloths, a white cotton cloth about 14" x 20", hemmed, to place materials upon; a piece of cloth 
or wash leather for wiping instruments; and a small dusting cloth. 

Box, to hold the pencils, erasers, cloths, and other small articles. 

5. Care and Arrangement of Equipment. Next in importance to having 
good instruments and materials is the necessity of handling them properly, 
keeping them clean, in good working condition, and in convenient, orderly arrange- 
ment. 



14 



ESSENTIALS OF MECHANICAL DRAFTING 



Keep the fingers clean, and the table and materials free from dust. The 
T-square and triangles especially will need frequent wiping. A paper or cloth may 
be fastened over a part of the drawing to protect it while working on other parts. 

Keep instrument case, box, and locker closed; and the ink bottle in the stand, 
with stopper in place. 

The instruments must always be carefully wiped, and properly replaced in 
the case when through working. 

Never permit ink to dry in the pens, or upon any part of the equipment. 

Take every precaution to insure against injury to the points, shanks, etc., of the 
instruments; and the edges of the scale, board, T-square, triangles, and curve 
rulers. 

Adjustments of joint-screws in compasses and dividers should be made by 
the instructor, unless otherwise directed. 

Inaccuracy, injury, or loss of any part of the equipment should be reported 
immediately. 

The table should be so placed that the light comes from the left* and, if 
possible, adjusted to such a height that the student may stand while at work. 

6. Drawing Board. The drawing board provides a flat surface upon which 
to secure the paper, and a st. edge against which to guide the head of the T-square. 

Either short edge may be selected for this 
purpose, but the board must be placed so 
that this edge will be at the left*, and no 
other used as a guiding edge in elemen- 
tary work. See Fig. 48. 

For convenience in working and to 
insure firmness and freedom in the use of 
T-square and triangles, the paper should 
be placed about 3" from the left* and 
lower edges of the board. 

Square the paper with the board by 
lining up one of its edges against the ruling 
edge of the T-square. Art. 7. 
Insert tacks about A" from each corner, pressing the paper as flat as possible 
with the hand; then with thumb or tack-driver force the heads flush with the 
paper, so that they will not interfere with the use of the T-square, etc. Always 
remove T-square from board when using tack-driver, and avoid marring the 
board. 

After paper is fastened, the board may be inclined by means of a book or 
block under its farther edge, so that all parts of the drawing may be more nearly 
at the same distance from the eyes. 

Keep paper secured to the board until drawing is completed. If removed 
before, it should be refitted by inserting a tack in one of its upper corners and 
lining up the most important horizontal of the drawing against the T-square. 

7. T-square. The T-square is used with its head against the left edge of 
the board. Fig. 48. The upper edge of the blade in this position is the ruling 

*In this and similar working directions, left-handed students may read "right" in place of "left."^ 




Fig 48 



INSTRUMENTS, MATERI.\LS, AND THEIR USE 



15 




Fig 49 



edge for all hor. lines, and guide for the triangles 
when drawing lines at certain Z s. Never use the 
lower edge of the blade as this would lead to errors 
difficult to trace. As the Z of the head and blade 
in different T-squares is apt to vary, the same T- 
square should be used until the drawing is com- 
pleted. 

(a) To DRAW A HORIZONTAL THROUGH A GIVEX 

POINT. Slide the head along the guiding edge of 

board until the ruling edge passes through the given 

pt. For preparation and use of pencil, see Art. 9. Move T-square by the head 

only, and while drawing the line keep head and blade securely in position by 

shding the fingers of the left hand along the blade and pressing towards the right. 

Fig. 48. 

8. Triangles. The triangles (Fig. 49) are used as rulers for Hues at Zs 
with the hor. direction. The ruHng edges of the 4-5° triangle form an Z of 90° 
and two of 45° each. Those of the 30° x 60° triangle form an Z of 90°, one of 
30°, and one of 60°. 

(a) To DRAW A LINE AT AN 
ANGLE OF 30°, 60°, 45°, OR 90°. 

Place one edge of the correspond- 
ing 30°, 60°, 45°, or 90° Z of the 
triangle against the T-square and 
guide the pencil along the other 
edge of the Z. Hold T-square 
and triangle firmly in position with 
left hand. Fig. 50. When the 
Une to be drawn is longer than 

the edge, shde the T-square until ^^■^X-^''''^ F'lG 50 

the required length is obtained. 

Avoid ruling near the corners, as thej' are apt to be rounded; thus, in drawing 
a line, say at 90° with a given hor., place the T-square a httle below the hor. as 
shown. Never rule against the inner edges. 

(b) To DRAW A LINE AT AN ANGLE OF 75° OR 

15°. Combine the triangles and T-square as in 
Fig. 51. The 30° Z added to one of 45° gives 
an Z of 75°. By reversing the upper triangle as 
shown by dotted hues, one of its edges will be at 
15°. Bj' placing the 60° angle against one of 45°, a 
line at 75° or 15° in the opposite direction may be 
obtained. 

(c) To DRAW A PARALLEL TO ANY GIVEN 

LINE.* Combine the triangles with an edge of one 
(see dotted triangle. Fig. 52) to coincide with the 

nATien the given line is hor.. or at .30°, 60°, 4.5°, 90°, 7.5°, or 1.5° with the hor. du-ection, the 
T-square, or triangle and T-square combinations, would ordinarily be used. 





16 



ESSENTIALS OF MECHANICAL DRAFTING 



given line A-B; tlien, liolding the second securely, 
slide the first until the edge which originally co- 
incided with A-B is in the required position C-D. 

(d) To DRAW A LINE AT 30°, 60°, 45°, 90°, 75°, 
OR 15° WITH ANY GIVEN LINE.* 

For 30°, 60°, 45°, or 90°, Fig. 53. Combine the 
triangles with an edge of one || to the given line 
A-B, as in (c). Then holding this triangle securely, 
shift the second, placing it against the || edge of 
the first, so that one of its edges makes the re^ 
quired Z with A-B. 

For 75° or 15°, Fig. 54. It is evident that after 
the second triangle has been placed against the || 
edge of the first, the first in turn must be shifted to 
give the recjuired Z . 

9. Pencils and Writing Pens. The hard 
pencil is used for ruling lines against the T-square, 
triangles, and curve rulers; the medium pencil for 
all freehand penciling, lettering, etc., and the pens 
for inking freehand lines, lettering, etc. 

(a) To SHARPEN THE PENCILS. Sharpen the 
end not bearing the grade stamp. Hold the inner 
side of both wrists firmly against the body, with 
the knife blade nearly flat against the upper side 
of the pencil and its cutting edge to the right. 
Cut with an outward wrist movement, removing 
the wood in long thin shavings, tapering it evenly 
down to but not cutting the lead, and so that about 
f " of the latter is exposed. Fig. 55. This method 
gives greater control of the knife and lessens the liability of soiling the fingers. The 
lead should be pointed by means of a lead-pointer. Hold pointer in the left hand, 
away from the table, and the pencil so that the entire length of lead will be tapered. 
Taper the lead of the freehand pencil to a fairly sharp pt. by rubbing it back 
and forth, at the same time rolling the pencil between thumb and fingers. Fig. (a). 

The lead of the ruling pencil should be tapered 
to a shai-p conic pt., or to a wedge pt., as the in- 
structor directs. The wedge pt. (Fig. b) retains its 
sharpness longer and fits more closely against the 
ruling edge. It is obtained by first tapering the 
lead slightly as for the freehand pt., and then 
rubbing the lead upon opposite sides to form a 
short, sharp edge at the end. 

Rub the leads frequently to keep them in 
proper condition. 

*When the given line is hor., or at 30°, 60°, 45°, 90°, 75°, or 15° with the hor. direction, the 
T-square, or triangle and T-square combinations, would ordinarily be used. 





Fig. 55 



INSTRUMENTS, MATERIALS. AND THEIR USE 



17 




Fig 56 



(b) Use of the Freehand Pencil. Hold 
pencil lightly, about I5 inches from the point. 
The relation of the pencil to the line, and the 
direction of the stroke, should usually be as in- 
dicated in Fig. 56. Face the paper squarely and 
avoid turning it while drawing. 

All lines should be drawn with as few strokes 
as possible. 

Draw lightly at first and correct any portion 
by drawing a second line before erasing. Finally 
strengthen the line to make it clear and even. 
Practice in arm and finger movements before 
drawing will aid in acquiring necessary freedom. 

(c) Use of the Ruling Pencil. Hold pen- 
cil as nearly upright as possible, with flat side of 
lead against the ruling edge. Steady pencil in 
this position by resting the tip of the third or 
fourth finger upon the ruler. Fig. 48. This per- 
mits the edge of the lead to wear evenly and give 
tmifoi-m lines. Bear lightly, — much pressure will 
dull the lead too rapidly, make uneven lines, or 
form depressions which cannot be erased. The 
result should be very fine, clear lines. 

The pencil should always be moved from left to right, the student turning his 
body or the board when necessary, that he maj' face the ruling edge squarely 
and do this more readily. Thus in drawing verts, face to the left, and draw 
awaj^ from the T-square. (See Fig. 50.) If ruled against a right-hand edge, the 
pencil is apt to glide away from the ruler and cause a break in the line. Watch 
the point constantlj^ as it is moved along. Never rule a line in a shadow, and 
never rule backward over a line. 

In drawing parallels move the ruxer from one position to the other in such 
manner that the preceding line and space will not be covered. Thus, in drawing 
II hors., move T-square downwards; in || verts., move triangle to the right. 

(d) Use of Writing Pens. Handle the pen the same as freehand pencil. 
Use little ink and aim to secure uniform, 
even lines. Exercise special care in work- 
ing over lines to avoid injuring the paper. 

10. Needle*point. The needle-point 
is used to set off distances from the scale, 
and to fix jjts. of Kne intersections which 
might otherwise be erased or lost. Hold 
needle upright and make the smallest 
puncture that can be seen. 

11. Scales. The scale is used for 
setting off measurements. It must never 




IS 



ESSENTIALS OF MECHANICAL DRAFTING 



be used as a ruler for drawing lines. The scales shown in Fig. 57 are graduated 
as described in Art. 4. 

(a) When the drawing is made so that each inch or fraction of an inch of 
measurement upon it is equal to the corresponding measurement on the object 
itself, the drawing is said to he full size or to a scale of 12" to afoot. 




Fig. 58 



When for convenience or necessity the drawing is made smaller or larger 
than full size each unit of measurement is made smaller or larger in proportion, — • 
thus when drawn, say Jialf size or 6" to afoot, each half inch on the drawing repre- 
sents one inch on the object; each quarter inch a half, and so on. 

Full size measurements are obtained from a scale graduated to full size inches 
and fractions. 

Half size measurements are usually obtained from a full size scale by 
simply reading each half inch on the scale as one inch, each quarter inch as a 
half, etc. Quarter, eighth, and sixteenth size can be obtained in like 




manner, 
to 1 



3" 



or froili scales graduated to 3", 1|", and 
ft. respectively. 

In scales graduated to feet and inches the first unit, 
when large enough, is divided to represent inches and 
fractions. For example, in the scale of IJ" to 1 ft. 
the first unit is divided into 12 eighth-inch parts to 
represent inches, and each of these subdivided to repre- 
sent halves and 4ths. Fig. 57. (b). 

When lOths, 20ths, 30ths, etc., of an inch are required, 
an engineer's or decimal scale is used. Any desired scale 
may be drawn by division of a line into the required pro- 
portional parts. See Art. 30. 

For method of determining the size or scale to be used, 
see Art. 100(c). 

(b) To SET OFF A MEASUREMENT (aS Say 2\"). 

Apply the scale directly to the line, with the "O" division 
exactly at the end of the line, and the needle-point at the 
2\" division. To set off, say 2 ft. and 2|", place the di- 
vision representing 2|" at one end of the line and a pt. at 
that representing 2 ft. 

Never transfer measurements from the scale with 
compasses or dividers. Successive measurements on a 



INSTRUMENTS, MATERIALS, AND THEIR USE 19 

line should, so far as possible, be set off without shifting the scale, so that an 
error in one distance may not affect all. 

12. Protractor. A Protractor (Fig. 58) is a scale used in laying off and 
measuring Z s. Its measuring edge is graduated to degrees and fractions (usually 
to half degrees), and the degrees numbered to read from to 180° in both direc- 
tions. 

(a) To DRAW A LINE AT ANY GIVEN ANGLE WITH A GIVEN LINE (say 40°). 

Place the protractor with its 180° line 0-0 against the given line A-B and center 
C at the given vertex A. Now place a pt., D, at the 40° mark. Remove the 
protractor and draw A-D, the required line. 

13. Compasses. The compass set (Fig. 59) is used for drawing circular 
■curves. The needle should first be adjusted as follows: — Release clamp screw 
A, and remove the lead holder. Insert the pe7i in place of the latter, and clamp 
securely. Then set the needle so that the point of its shouldered end will be even 
with the pen point. Once set, the needle should not be changed. The lead 
only will need resetting, as it wears away. 

(a) To PREPARE THE COMPASSES FOR PENCILING. 

Place a 4H compass lead, about 1" long, in the holder, 
and taper it as directed for the ruling pencil. Refit 
the holder and adjust the lead to the length of the 
needle, with its edge so placed that a fine, even line 
will result when the compasses are revolved. 

(b) To DRAW A CIRCLE. Open the compasses and 
adjust the legs at the joints so that both lead and 
needle will be at right Z s to the paper while drawing. 
This is necessaiy to prevent the lead from wearing un- 
evenly and the needle from digging into the paper. 
The puncture should be barely visible. IG.DU 

Hold the compasses by the handle only, with thumb and first two fingers 
(see Fig. 60) and always revolve it around to the right (clockwise). Bearing 
lightly and evenly upon the lead point, draw the curve with one continuous 
motion, stopping exactly at the end of the revolution to avoid widening the line. 

In drawing a O or arc of given rad., first set off the rad. upon a C.L., and, in 
placing the compass point at the center, steady the needle with a finger of the 
left hand. 

Changes of rad. should be made, so far as possible, with the right hand only, 
and care taken not to enlarge the center. 

(c) To INK CIRCLES AND CIRCULAR ARCS. Insert the pen and clamp it 
securely. Clean, fill, and set the pen, as directed in Art. 17(a). Open compasses 
to the rad. of the penciled curve and adjust the legs, as directed in (b). To give 
clean-cut lines, both blades must bear evenly upon the paper. The directions 
for penciHng apply also to inking. Before placing the pen point upon the curve, 
the compasses should first be revolved over the line, in space, to make sure that 
the inked line will pass exactly through the desired pts. Errors caused by enlarge- 
ment of centers may thus be avoided. Do not go over a line a second time. 




20 



ESSENTIALS OF MECHANICAL DRAFTINCx 




Fig. 6 



Fig 62 



(d) Lengthening Bar. When the rad. is too great to admit of placing the 
points _L to the paper, the lead or pen leg should be extended by means of the 
lengthening bar. In this case, the needle leg may be steadied with the left hand 
and the drawing point moved with the other, care being taken not to change the rad. 

14. Bow Compasses. The boio pencil (Fig. 
61) and bow pen (Fig- 62) are used for drawing 
small Os and arcs for which the large com- 
passes are not convenient. Do not use them 
for radii over f". The directions for prepar-' 
ing and using the bow compasses are much 
the same as for the large compasses. The 
needle must project slightly beyond the pen or 
lead point, and the lead be tapered more neai-ty 
to a pt. To enable the points to be brought 
close together, the needle is generally flattened 
on its inner side. The rad. is adjusted by 
turning the thumb-nut on the connecting bar. 

Before turning, spring the points together so 
that the wear of the screw thread may be 
lessened and the adjustment made more readily. 

15. Dividers. The dividers or spacers (Fig. 63) are used for transferring 
measurements fr'om one part of a drawing to another, and for setting off equal 
distances on a line when they cannot readily be laid off by means of the scale. 

I Handle the dividers in the same general way as the compasses. 

In some dividers one leg is furnished with a hairspring and nut by 
means of which this leg may be moved for slight changes of adjustment. 

(a) To DIVIDE A STRAIGHT LINE OR A CIRCULAR CURVE INTO 

ANY NUMBER OF EQUAL PARTS (say 3). Opeii the dividers to' a dis- 
tance equal (by ej'e) to ^ of the line to be divided, place one of its points 
upon the end of the line and revolve the dividers until the other point 
is exactly on the line. Proceed in this manner, revolving alternately 
in opposite directions, until the distance taken has been set off three 
times. ' Never remove both points at the same time. If the -distance 
taken does not apply exactly it must be increased or diminished by 
an amount equal to i of the difference, and the trial repeated until the 
line is equally divided. No pts. should be made visible until the 
divisions have been verified as correct. 

16. Bow Dividers. Fig. 64. These are used for small distances, 
and in the same general manner as the large dividers. The points are 
adjusted as directed for bow compasses, Art. 14. 

17. Ruling Pen. The ruling pen (Fig. 65) is used for inking all 
, lines other than circular curves. 

(a) To FILL AND CLEAN THE PEN. Before filling the pen, moisten 
a folded end of the penwiper and draw it gently between the blades. 
Fig. 63 When clean and dry bring the blades together at the point by means 



INSTRUMENTS, MATERIALS, AND THEIR USE 



21 




Fig. 64 



Fig. 65 



of the thumb-screw; then holding the pen upright, 
not over the drawing, insert the ink between the 
blades with the filler. Do not fill above J" from 
the point, otherwise the ink will flow out too freely. 
See also that there is no ink on the outside of the 
point, as this will widen the line, make it ragged, or 
cause a blot. Replace the stopper immediately to 
prevent the ink from thickening. 

Having filled the pen, set the blades to the re- 
quired width of line. Always try the pen on a piece 
of waste paper before using it on the drawing. To 
insure its flowing freely, the amount of ink in the pen 
must be kept as nearly as possible the same. Avoid 
having to piece out a line. As the ink dries rapidly, 
the pen must be cleaned and refilled frequently. In 
doing this, it is not necessary to open the blades. 
The setting should remain unchanged until all lines of the same 
width are inked. If the pen fails to work, it should be sharpened 
by an experienced person. Never put the pen aside without care- 
fully cleaning it. 

(b) To RULE A LINE. Hold and steady the pen as directed 
for the pencil (Art. 9 (c)), the first finger resting on the flat side of 
the pen above the thumb-screw and the second against the edges of 
the blades as shown in Fig. 50. 

Adjustments for width are made with thumb and second finger of the same hand. 

Guard against getting the pen point too close to the ruhng edge by placing 
the latter slightly away from the Hne, as shown. A slight downward pressure 
only should be necessary, but the points of both blades must bear evenly upon 
the paper to give a clean-cut Une. Bear lightly against the ruling 
edge to prevent varying the width of the line. Always steady the 
hand, and move the pen from left to right as directed for penciling. 
Just before reaching the end of the line stop the arm movement 
and complete the line with a finger movement, then lift the pen 
immediately and move the ruling edge away from the line. 

In ruling curves, turn the pen gradually so that the blades will 
not be at an Z with the ruhng edge. Art. 18. Never use the ruHng 
pen freehand. 

18. Curve Rulers. These are used for ruhng curves that 
cannot be drawn with the compasses. They are made in various 
shapes and sizes. Two of the most serviceable are shown in 
Figs. 66, 67. 

(a) To HTJLE A CURVE, ABC. Fig. 67. First sketch the 
curve hghtly with freehand pencil, through previously determined 
pts. Now find, by trial, a portion of the ruler which will fit as 
much of the curve as can be ruled conveniently at onetime, as A-B, 
and true up that part by tracing over it with the ruling pencil. 




Fig. 66 



22 



ESSENTIALS OF MECHANICAL DRAFTING 



Match succeeding parts in the same manner, making the edge fit over a portion 
of each preceding part to insure an even, unbroken line. The freehand pencihng 
should not be omitted, as the tendency would be to make the ruled line curve out 
too much or too little. Having trued the line, rub the soft rubber lightly over it. 

In inking the curve, use the ruling pen as described in Art. 17(b). In ruling 
curves symmetrical about one or more axes, as ellipses, helices, etc., the portion 
of the ruler used for one part should be noted and used for corresponding parts. 
Sharp turns at ends of axes should first be drawn by means of compasses, the 
centers being taken in the axes and care taken to use the proper rad. and length 
of arc. 

(b) Non-circular curves may often be approximated throughout by tangent 
arcs. Thus, in inking the curve shown by the fine line in Fig. 68, beginning say 
at A, determine by trial the center and rad. of as much of an arc as will practically 
coincide with the curve. Ink this arc; then, changing the center and rad., ink 
the next portion; note that the centers must be on the line through the pt. of 
tangency. 




Fig. 67 




Fig. 68 



IQ. Erasers. (a) The soft rubber is used for pencil erasing and paper 
cleaning. Keep paper as clean as possible from the start. (See Art. 5.) The 
softest rubber is liable to roughen the paper, making it difficult to keep clean 
and to obtain sharp lines. In removing a line, rub lengthwise. Avoid much 
pressure and always remove dust before proceeding with drawing. When a 
drawing is finished in ink, the eraser may, if necessary, be passed lightly over the 
entire surface, care being taken to avoid dulling the lines. 

(b) The hard rubber is used for ink erasing. The part to be remoyed should 
first be allowed to dry. Care must be taken not to injure the surface. Never 
use a knife. If the surface is roughened by erasing, smooth it as well as possible 
with the finger nail. A thin card or piece of celluloid with narrow openings can 
be used to protect adjacent parts when erasing. 



CHAPTER III 
PENCILING AND FINISH RENDERING 

20. Layout of the Sheet, (a) Margin Lines. To improve the general 
appearance of the drawing and to insure keeping all lines and figures a safe distance 
from the edges of the sheet, it is customary to rule border lines with uniform 
marginal spaces at top, bottom, and sides. See Fig. 69. 

Assuming the paper to be ll"xlo", the required border 10"xl3|", and 
marginal spaces §", proceed to lay out the border and trim lines as follows: 
Having tacked the paper to the board (Art.G) setoff two pts. near the left edge, 
I" and 10|" respectively, above the lower edge, Art. 11. Through these pts. 
draw Kght horizontals across the sheet, Art. 7. On the lower hor., set off a pt. 
I" from the left edge; |" from this place a second; Idh" from the second, a 
third; and |" from that a fourth. Through these pts., draw the 90° lines 
(verticals), Art. 8. Test distances between verts, at the top and hors. at the 
right with the scale. 




Fig. 69 



When the sheet is completed and removed from the board, the J" strips con- 
taining the tack holes are to be cut off, thus leaving the sheet 11" x 14V'. 

(b) Loc.\TioN OF Name, Title, etc. On elementary drawings these maj' 
be lettered as shown. For titles, etc., on working drawings, see Art. 102. For 
instructions in lettering, see Art. 25. 

21. Constructive Stage of the Drawing. All lines should be penciled first 
in uniform, fine full lines, as indicated in Fig. 209(a), (b), (c). Then, to avoid 
errors in the finishing stage (Art. 22), lines representing edges and outlines of the 
object should be gone over with a slightlj' firmer pressui-e, and the hidden parts 

23 



24 ESSENTIALS OF MECHANICAL DRAFTING 

dashed as in Fig.(d). The general order of pencihng different kinds of drawings 
is indicated in Art. 3. 

To avoid the necessity of piecing out, make the hnes first of indefinite length. 
Be sure that all measurements are set off upon definite lines and that lines intended 
to pass through particular pts. actually do so. No part of the penciling should 
be slighted. Inaccuracies can seldom be corrected in the process of finishing. 
Aim to do no erasing until the penciling is completed. 

To secure greater accuracy and economy of time, similar operations should 
be grouped. Thus, draw all lines that can be ruled with the T-square and triangles 
in one position at the same time, and ||s of one set before commencing those o£ 
another. When using the scale or dividers, set off all distances possible at once. 
When using the compasses work the circular curves in the same way, drawing 
those having the same rad. at one setting of the instrument, etc. As st. lines 
can be drawn tangent to curves more accurately than the reverse, pencil the curves 
first whenever possible. 

To enable "the method of procedure to be readily followed, all lines used in 
making the drawing should be left upon the sheet until it has been approved. 

22. Finishing Stage of the Drawing. For greater distinctness and per- 
manence the drawing is usually lined-in or finished with ink, either by going over 
the lines on the original or by making a tracing as in Art. 106. Drawings not 
intended for continued use or of which no copies are needed are often finished in 
pencil. 

Do not commence the finishing stage until the constructive stage is completed 
and approved. See that the drawing surface is free from dust. 

(a) Inking. Be careful to make sharp, even lines, and see that all lines 
begin and end exactly where it is intended that they should. To prevent lines 
running together at their intersection, see that the first is thoroughly dry before 
inking the second. Do not use a blotter. 

(b) Finishing in Pencil. If the drawing is to be finished in pencil, the aim 
should be to secure as nearly as possible the accuracy and distinctness of an inked 
drawing. The medium pencil should be used, at least for strengthening object 
lines. In finishing dashed lines it is not necessary to era,se lines of the constructive 
stage between dashes as they will not be prominent if the finish lines are properly 
emphasized. 

(c) Line Conventions. The different purposes of the lines of the drawing 
are indicated by varying their character, width, or color. The conventions shown 
in Fig. 70 are commonly used on drawings finished wholly in black ink or in 
pencil, for the purposes stated below. They are suitable in width, length of 
dash, etc., for ordinary drawings. 

A — Visible lines of objects in all required views* and edges in developments. 

B — Hidden lines of objects in all required views. As a rule end dashes should 
touch the limiting lines. 

C — Visible lines of objects in auxiliary views used in determining required views. 
When aux. views show visible lines only, they may be finished as construction lines. 

*When shadow or shade lines are used, shade the curved edges as each is lined-in. (See Art. 
23(c).) St. shadow lines are left in pencil until all but the border is finished. 



PENCILING AND FINISH RENDERING 25 

D — Hidden lines of objects in auxiliary views used in determining required views. 

E — Traces of projection, section, and base planes; center lines, and axes. Dashes 
of section traces, center lines and axes should extend about |" beyond the view 
or part on which they are drawn. 

F — Construction {loorking) lines required to shoiu method of construction; pro- 
jectors, and extension lines. Dashes of extension lines should not touch lines 
of views, and should extend about -nr" beyond arrowheads of dimension lines. 

G — Dimension lines, and pointers. 

H — Break lines. Same as edges, etc., but should be drawn freehand. 

I — Arroivheads, figures, notes, titles, etc. These should be penciled freehand, 
and in inked drawings always finished in black. See Fig. 74. 

J — Section lining. Same as C, or as in Fig. 187. 

K — Line shading. As in Art. 24. 

L — Straight shadoiv and shade lines of all required views, and border lines. 

Red ink. In inked drawings, the use of full red lines for all Hues under C, E, 
F, and G, and dashed red for those under D is more economical of time and makes 
the required views more prominent. 



Width 2 



B 

C 

D 

E • 

F 

I I " 

G h 3t 



2 



L 



Fig. 70 



(d) Order of Finishing. It is desirable to complete all lines and parts of 
similar character before proceeding with those of another, in the general order 
indicated in Art.(c), in observing which, similar operations should be grouped 
as in the following : — 

Circles and arcs, beginning with the smallest; non-circular curves; horizontals, 
beginning with those at the top ; verticals, beginning with those at the left ; obliques, 
beginning with those obtainable by T-square and triangle combinations. 

Do not use a triangle alone, unless necessary. 

In finishing break lines, arrowheads, figures, and notes, etc., work from upper 
part of sheet downward. 

23. Shadow Lining. In practical drawings, visible edges and outlines of 
objects are generally indicated by full lines of uniform width. It is sometimes 
desirable, however, to finish certain lines wider than the others for the purpose of 
giving an appearance of relief to the drawing and to indicate the relative positions 
of the surfaces more clearly. 



26 



ESSENTIALS OF MECHANICAL DRAFTING 




conventional shadow lines 
Fig. 71 



PENCILING AND FINISH RENDERING 27 

(a) If a cube placed as in Fig. 71(a) be lighted by || rays coming over the 
left shoulder, in the direction of diagonal A-B, it is evident that the upper, left, 
and front faces will be in the light, the others in the shade. 

Lines separating light from dark surfaces are called shade lines. The visible 
shade lines of the cube will, therefore, be the lower and right lines in the front view, 
and the upper and right lines in the top view, and may be indicated by broad 
lines as shown. 

Assuming the same direction for the light, the visible shade lines in all 
rectangular objects and parts, whose axes are || or J. to the planes of the views, 
would have the same general locations as in the cube. In such cases the shade 
lines can usually be determined by eye. The determination of the actual shade 
lines in all cases, however, would involve considerable time and labor, and their 
locations would frequently be such as to complicate the drawing rather than 
aid in explaining the form of the object; hence it is customary to disregard the 
actual shades and shadows and to apply the broad lines in such manner that 
they will indicate edges separating visible from hidden surfaces only, and so as 
to produce an effect of narrow shadows cast by the object. 

It is convenient to regard these lines as shadow lines rather than as shade lines. 

To further simplify the application, most draftsmen shade or shadow-line 
all views the same as the front. 

The aid given by such lines in reading a drawing will be apparent from the 
figures. 

The directions for this conventional method may be stated as follows: — 

(b) Shadow-line the lower and right lines of intersection between visible and 
hidden surfaces in all views, regardless of the actual shades or shadows of the 
object. This includes both sharp and slightly rounded edges. Lines not repre- 
senting such edges are generally shadow-lined only when surfaces extend back 
from them at right Zs to the plane of the drawing, in order to indicate more 
clearly that they do not represent edges. A sectional view is generally shadow- 
lined just as if the portion shown were complete. Lines representing broken 
surfaces are preferably not shadow-lined. Never shadow-line the line of inter- 
section between visible surfaces, nor dashed lines. 

(c) To SHADOW-LINE A DRAWING. In determining the shadow lines remember 
that the rays of light will be at 45° down to the right in all views and that all 
views are shadow-lined in the same manner. Limiting rays may be penciled as 
shown. Each edge to be shadow-lined should be indicated by a mark upon the 
line before inking is begun. Drawings are never shadow-lined in pencil. 

Straight Lines. A st. shadow line should be ruled the required width (3) at 
one stroke of the ruling pen, and its width added on the outside of the pen- 
cil line. 

Circular Curves. First ink the curve in the usual manner (width 2). Then 
taking a second center below and to the right of the first, on a 45° line, at a dis- 
tance equal to width 3 and without change of rad. or setting of pen, draw an arc 
on the outside or inside of the first curve according to the required location of 
the shadow portion. (See Fig. 71(h).) If an uninked portion remains between 
the arcs, spring the instrument slightly to fill it in. 



28 



ESSENTIALS OF MECHANICAL DRAFTING 



Irregular Curves. First ink the widest portion; then with the ruling pen set 
to width 2, blend this carefully into the rest of the curve. 

24. Line Shading. This is a conventional method of producing an effect 
of light and shade, corresponding to that upon the object itself by means of lines, 
usually of graded widths and spacing. Fig. 72. It is used on drawings intended 
for illustrative purposes and in cases where it is necessary to indicate the direction 
of certain surfaces more clearly than would be possible by the mere outline or by 
shadow lines. 




Fig 72 



(a) The direction of the light is assumed as for shadow-lining (Art. 23) ; the 
same for all views. The right and lower sides are, therefore, the dark sides on 
convex surfaces and the left and upper on concave surfaces. In regular curved 
surfaces the darkest portion is about | of the rad. from the center. It may be 
accurately determined as shown in top view of Fig. (a). The spacing may be 
indicated in pencil, as below Fig. (a). Shade first the dark portion, beginning 
at about j of the rad. from the center. Use fine lines only on the light side, 
somewhat farther apart than on the dark, and stop at about f of the rad. from 
the center. On small parts shade dark side only. (See Fig.(g).) Figs, (a) and (i) 



PENCILING AND FINISH RENDERING 29 

illustrate methods of shading fillets, beads, etc., on large drawings; Figs.(e), 
(f); (g), for small drawings. The shading of conic surfaces by |1 lines, as shown, 
is less difficult and generally more satisfactor}^ than by radial lines. When 
desirable to shade plane surfaces, the method in Fig.(d) may be used. 

25. Lettering. The names, titles, notes, and dimensions required on drawings 
should be carefully lettered in even, well-proportioned, and well-spaced characters. 

The styles shown in Fig. 73 are those most generally used. 

The inclined Gothic differs from the vertical only in the slant. The ^-ertical 
is sometimes chosen for titles and headings, and the inclined for notes and dimen- 
sions. A uniform style for all lettering is more common. 

The capitals may be used with or without lower-case (small) letters, or with small 
capitals in place of the latter. Titles and headings are usually more satisfactory 
if all capitals are used, while notes are more easily read if composed of capitals 
and lower-case letters. 

(a) Peoportions. The character, proportions, and relations of the ele- 
ments composing the letters and numerals should be carefuUj' noted and fixed 
in mind. Examination of the vertical Gothic will show that W is the widest 
character, M next, and A, V, and 4 next; that I, J, and 1 are narrowest; and that 
the others are nearly equal in width to letter 0. For general purposes this 
width should be from | to | of the height. These variations are designed to 
overcome the appearance of inequalitj^ which would result if all were made equal 
when used in words. For hke reason A is extended slightly above the other 
letters and V below. The curves of C, G, J, 0, Q, S, U, 2, 3, 5, 6, 8, and 9 also 
extend slightly beyond. To avoid the effect of top-heaviness the upper part of 
B, C, E, G, K, R, S, X, Z, 2, 3, 5, 6, and 8 is slightly narrower than the lower, 
and certain parts in B, E, F, H, R, S, X, 3, 5, 6, and 8 come slightly above the 
middle, while in A, G, K, P, Y, 4, and 9, they come below. All curves are elliptic. 

In practice the variations in height and width noted above should be estimated 
by eye. 

The body of the lower-case letters should be f or | of the initial capitals. 
The sizes of letters and numerals generally suitable are indicated in Fig. 74. 

(b) Spacing. The spaces between letters vary in shape with each different 
combination. In order to make these spaces appear equal in size and thus avoid 
the effect of crowding or isolating letters, it is necessary to increase the spaces 
in certain cases and decrease them in others. Thus, considering the vertical 
capitals, the spaces of I should be considerably greater than of other letters, 
especially when parts of adjoining letters are 1| to it. In letters whose sides are 
curved as in 0, C, D, etc., it is generally necessary to decrease the space. This is 
also true of letters having oblique sides as in A, V, etc., and letters having a 
greater space at one or both sides as in L, J, P, T, etc. 

The simplest method of obtaining good spacing is to sketch the words lightly 
and study the effect. 

The space between words should be about Ij times the M-idth of letter O; 
that between sentences in line about three O's; and that between || lines of letter- 
ing about equal to the height of the shortest letters in either. 



30 ESSENTIALS OF MECHANICAL DRAFTING 

(c) Penciling. In penciling titles, etc., and notes, rule only the hor. guide 
lines, and a few vert, or slant lines to preserve the correct positions. For method 
of planning a title see Art. 102(c). In dimensioning drawings estimate the 
heights of the numerals by eye. 

Draw all characters freehand, using a fairly sharp pencil point. Art. 9. 

Before lettering a drawing, the style to be used should be practiced until 
it can be done reasonably well. Unless otherwise directed, begin with the vertical 
capitals, observing the order given by the numbers below them; then practice 
word-combinations; then the figures. 

Place Fig. 73 close to the work and analyze each character. A convenient 
order for drawing the strokes is indicated by the arrows. To determine the proper 
proportions and spacing, first point the forms and sketch lightly the main lines. 

Bear in mind that the value of the practice is in the carefulness and not in the 
amount. Do not mix the styles. 

(d) Inking. Use the medium-point writing pen for small letters and 
figures, and the coarse pen for large letters, etc. The width of the lines should 
be uniform and obtained at one stroke. 

Observe number, order, and direction of strokes as in penciling. 






M||l| 









i*i 



o^mo 



^/if^ 



S3C 






s:(-'^o 



< 

(J 

> 



o '— m pn. ^ry 

^-^ ^^^M ^^^" ^^^^ 






^^ i^T*i .^^ i<n 



i-r^ ^^ 1*^ 



>sL^ 






1=^ 



^tr>tt 



I 
X 



R 

I I 



I [ 

id 

I I , 



1 1 1 



Li 







o 




f^ 



1^ 



o 
o 

Q 
U 



! I 



m 



L n 



^1 



o 




U 



I ! I 



, I I 

! N.I 



w 




I I 






! I 



Pfl 



I I 
^1 



"'f 



u. 






I 



^ 



'A 

fj3 



Fis.73 



n 




TITLE AND FILING INDEX. 



SPACE ABOUT 2X4-5: 



^ SPEED LATHE 

HEAD STOCK DETAILS 

SMITH MFG CO. NEW YORK 
Scale: Full Size Date: 4-7-18 

Dr. A.R5. Jr. V. L. Ch. H.M. App. WFR.. 



S.L:IO-5 





BILL OF MATERIAL. 










:Na 


NAME 


REQ'D 


MAT. 


REMARKS 




I 1 


Plain Bearing 


1 


C.I. 


Patt.#B-5 


T 


L2 


St'd Hex. Bolts 5x2" 


2 


W.I. 


C.H.H'ds &Nuts 




U-3-*- 


< ^1 , 


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7 






r 8 


^4. 


h" 2 - 


a * 








t -|'0 = 

-l<o t 



j -cor 
- <o t 
T -i<o 



SUB TITLES AND IDENTIFYING MARK. 

Face Plate S.LrS 

Scale: Half Size 
Make 2 C.I. Po+t. *F-4 

f:a.o. 



NAMES OF VIEWS. SECTION PLANES. ETC. 



^SECTION 



PLAN DIAGRAM A- 



■B 



NOTES, FINISH MARKS AND DIMENSIONS. 



-t^^ ± Tap |3Th. 4-HolesJor M-6. fxfKey 
I. -i- ^ v/* I . ,5" |=-i«r ,„B ore. 1 1 3"r^ 



H 



Fig. 74 



32 



CHAPTER IV 



GEOMETRIC CONSTRUCTION 



26. Geometric and Practical Methods. The accurate execution of the 
vie-\vs and diagrams by which the hnes and surfaces of an object are represented 
involves the graphic solution of various plane problems such as the division of 
lines and the construction of perpendiculars, parallels, polygons, etc. 

In geometry (Art. 2) these solutions are obtained by the orderly application 
of geometric principles, which require the use of the compasses and a straightedge 
only in dra^ndng the necessary working or construction lines. 

While it is frequently necessary in practical drafting to obtain the solution 
bj- a geometric method, the draftsman is generally enabled to shorten the con- 
structive process or to obtain a direct solution by means of the T-square, triangles, 
dividers, etc. 

The constructions explained in this Chap, are those most commonly apphed 
not only in drafting, but in the laying out of the actual lines and surfaces of 
objects by the workman. 

The geometric method is that first given. Where a practical method (P. M.) 
is not also given, the geometric method would ordinarily be used. 

27. To bisect a straight line, A-B, or a circular 
arc, A4B. Fig. 75. With A and B as centers and 
any rad. describe arcs intersecting in pts. 1 and 2. 
Through 1 and 2 draw a st. line, which will be J_ 
to A-B and bisect it at 3, and the arc A4B at 4. 

Xote 1. — Every pt. in the J_ bisector 1-2 is equidistant from 
A and B; hence a different rad. ma^' be taken for each of its 
determining pts. 1 and 2, or both located upon the same side of 
A-B. 

Note 2. — The ± bisector of a chord, if extended, will pass 
through the center, 5, of the O of which the arc A4B is a part, 
and will bisect the O , also the Z AoB of which the arc A-B is 
the measure. 

Note 3. — The method of drawing a O or an arc of given rad. 
through two given pts., or upon a given chord, is evident. 

P.M. Obtain division with the dividers. Art. 1.5(a). 

28. To bisect an angle, CAB. Fig. 76. With 
any rad. locate pts. 1 and 2 upon the sides, equi- 
distant from vertex A. With 1 and 2 as centers, 
any rad., locate an equidistant pt. 3. Draw 3-A, the 
bisector of the Z . 

Note. — When the vertex, A, is not usable, draw ||s to the 
given sides (see Art. 36), obtaining A', and bisect Z C'A'B'. 

33 




Fig 76 



34 



ESSENTIALS OF MECHANICAL DRAFTING 



29. To construct an angle equal to a given angle, CAB. Fig. 77. Draw an 
indefinite line A'-B'. With A as center and any rad., draw arc C-B to cut the 
sides of the given Z. . With A' as center, same rad., draw an indefinite arc C'-B'. 
With the chord of C-B as rad. and center B', cut C'-B', at C. Draw A'-C com- 
pleting the required Z . 

30. To divide a straight line, A-B, into any number of equal parts (say 5). 
Fig. 78. Draw A-4 at any Z with A-B. Draw B-1' at the same Z, Art. 29. 
From A and B set off any distance upon A-4 and B-1', as many times as the 
required number of divisions on A-B, less one. Draw 4-4', 3-3', etc., which will 
divide A-B as required. 

Second Method. Set off five equal distances 
upon A-5 from A, and draw ||s to 5-B by making Zs 
at 4, 3, 2, and 1 equal to Z A5B. 

Note. — Any line, as 1-1", || to one side, 5-B, of a A, divides 
the other two sides, A-B and A-5, proportionally. Thus the 
ratio of A-1" to 1"-B equals the ratio of A-1 to 1-5; also A-1" 
is to A-1 as A-B is to A-5. 

P. M. Draw A-5 at any Z with A-B and from 
A set off five equal distances upon it. Draw ||s to 
5-B through 4, 3, 2 and 1 by Art. 8(c), dividing 
A-B as required; or obtain divisions by Art. 15(a). 

31. To divide a straight line, A-E, into parts 
proportional to those of a given divided line, A-D. 

Fig. 79. The method is evident from Art. 30. 




32. To lay off the length of a given circular curve 
upon a straight line. There is no exact method. 
Divide the given curve into short arcs, not neces- 
sarily equal, whose chords will closely approximate 
the lengths of the subtended arcs. Lay off these 
chords successively upon the st. line. 

Note. — The length of the circumference of a O is 3.1416 
times the diam.; hence it may be computed and the nearest 
fraction taken from a table of decimal equivalents. 

33. To lay off the length of a given straight line 
upon an arc. The method is evident from Art. 32. 

34. To draw a perpendicular to a line, A-B, 
from or through a given point C. 

(a) When C is upon A-B, at or near the middle of the line. Fig. 80. 
Regard A-B as a st. Z with vertex at C and proceed as in Art. 28. C-3 will be 
the required _L. 

(b) When C is opposite or nearly opposite the middle of A-B. 
Locate pts. 1 and 2 equidistant from C, and pt. 3 equidistant from 
as in (a) . Draw C-3, the required _L . 

(c) When C is upon A-B and at or near the end of the line. 
With C as center, any rad., draw an arc cutting A-B at 1. With 1 



Fig. 79 



Fig. 81. 
1 and 2, 



Fig. 82. 
as center, 



GEOMETRIC CONSTRUCTION 



35 



same rad., cut this arc at 2. With 2 as centei-, same 
rad., draw an arc above C. Through 1 and 2 draw 
a St. line to cut the last arc at 3. Draw C-3, the 
required ±. 

Note 1 . — The dotted arcs suggest another method of locat- 
ing pt. 3. 

Note 2. — Any Z,as 1C3, inscribed in a semicircle, is aright Z. 

Second Method. Fig. 82. Assume any pt. 2, 
not upon A-B, as center, and with rad. 2-C, draw an 
arc cutting A-B at 1. Through 1 and 2 draw a st. 
line to cut this arc at 3. Draw C-3. 

(d) When C is opposite or nearly opposite 
THE END OF A-B. Fig. 83. From C draw any line 
C-1, intersecting A-B obliquely. Bisect C-1 at 2. 
With 2 as center, rad. 2-C, cut A-B at 3. Draw 3-C, 
the required J.. 

Note. — Compare with second method of preceding. 

Second Method. Fig. 83. From any two pts. 
on A-B, as 1 and A, and radii 1-C and A-C, describe 
arcs to intersect «,t 4. Draw C-4 . 

P. M. For all cases: Obtain the J. by Art. 8(d). 

35. To draw a line at an angle of any given 
magnitude in a quadrant, with a given line, A-B, 
at A. Fig. 84. 

(a) 45°. Draw A-1 at 90° with A-B by 
Art. 34(c). Bisect Z lAB by Art. 28; then A-3 will 
make an Z of 45° with A-B. See also Note 1. 

(b) 60°. With A as center, any rad., draw an 
arc cutting A-B at 2. With 2 as center, same rad., 
cut this arc at 4. Draw A-4, which will make an Z 
of 60° with A-B. 

a 60° Z 4AB as in (b) 
5 will be at 30° with A-B. 



(c) 30°. Determine 
and bisect it; then line A 
See also Note 1. 

(d) 15°. 
and bisect it 

(e) 75°. 
and a 60° Z 
1A4 and A-7 will be at 75 

(f) By trisecting the 



(c) 



Determine a 30° Z 5AB, as in 
then A-6 will be at 15° with A-B. 
Determine a 90° Z lAB as in (a) 
4AB as in (b). Bisect the 30° Z 

with A B. 

15° arcs of quadrant 1-2 



and dividing the 5° arcs thus obtained into degrees, 
by trial (Art. 15(a)), a line at any intermediate 
degree may be determined. Lines at Zs involving 
fractions of degrees, may be determined by the same 
general methods. 




Fig 




Fig 




Fig 82 



'\ 




Fig. 




Fig. 



Note 1. — The dotted lines suggest another method of locating pts. 3 and 5. 

Note 2. — Observe that the rad. of a O is equal to a chord of ^ of its circumference 



36 



ESSENTIALS OF MECHANICAL DRAFTING 





Fig 65 




Fig 87 





P. M. Lines at 90°, 45°, 60°, 30°, 15°, and 75* 
may be obtained by Art. 8(d). For lines at inter- 
mediate degrees see (f). The protractor (Art. 12) 
may be used for all cases not involving fractions 



Fig. 89 



other than 1°. 

36. To draw a parallel to a given straight line, 
A-B. 

(a) At a given distance, C-D, from A-B. 
Fig. 85. With any two pts., 1 and 2, on A-B as 
centers and C-D as rad., describe arcs on the same 
side of A-B. At 1 draw a ± to A-B, cutting the arc 
from 1 at 3. With 3 as center and rad. 1-2, cut the 
arc from 2 at 4. Draw 3-4, the required 1 1 . 

P. M. At any pt., 1, on A-B draw a ± 1-3 (Art. 
8(d)), and make 1-3 equal to C-D. Through 3 draw 
3-4 II to A-B. Art. 8(c). 

(b) Through a given point 3. Fig. 85. Lo- 
cate any two pts., 1 and 2, on A-B. With 1-3 as 
rad. and pt. 2 as center describe an arc on the same 
side of A-B as pt. 3. With 3 as caiter and rad. 1-2 
cut the last arc at 4. Draw the required parallel 3-4. 

P.M. Draw 3-4 II to A-B by Art. 8(c). 

37. To construct a triangle. 

(a) When the sides are given. Fig. 86. 
Make A-B equal to one of the given sides. With A 
and B as centers and radii equal to the second and 
third sides draw arcs intersecting at C. Draw A-C 
and B-C to complete the A. 

P. M. When the given sides are equal draw the 
second and third at 60° with the first by Art. 8(d). 

(b) When a side A-B, and the angles at A 
and B, are given. Fig. 86. Make the side A-B 
and Zs CAB and ABC equal to the given side and 
Z s, and extend A-C and B-C to meet at C. 

(c) When a side A-B, the angle at B, and 
the angle opposite A-B, are given. Fig. 86. 
Find the angular magnitude at A by subtracting 
the sum of the known Zs at B and C from 180° and 
proceed as in (b). See Fig. 20. 

38. To construct an isosceles triangle when the 
base, A-B, and vertex angle, ACB, are given. Fig. 86. 
Find the angular magnitudes at A and B by subtract- 
ing the known Z from 180° and bisecting the remain- 
der; then proceed as in Art. 37(b). 



GEOMETRIC CONSTRUCTION 



37 



39. To construct an equilateral triangle when 
the altitude, A-B, is given. Fig. 87. Draw B-1 and 
B-2 at 30° with A-B, Art. 35(c). Draw 1-2 ± to 
A-B, Art. 34(c). 

P. M. Draw B-1 and B-2 at 30° with A-B, 
Art. 8(d). By same Art. draw 1-2 J. to A-B. 

40. To circumscribe a circle about a given 
triangle, ABC. Fig. 88(a). Draw the ± bisectors 
of either two sides, as A-B and B-C (Art. 27) to 
intersect at 1; pt. 1, being equidistant from A, B, and 
C, will be the center of the required O . 

Note 1. — The center of any O is at the intersection of the 
± bisectors of any two of its non-parallel chords. 

Note 2. — The method of drawing a O through any three pts., 
not in the same st. line, is evident. 

41. To inscribe a circle within a given triangle, 
ABC. Fig. 89. Draw the bisectors of either two of 
the Zs, as CAB and ABC (Art. 28) to intersect at 1, 
the center of the required O . The _L distance from 
the center to any side is its rad. 

42. To inscribe an equilateral triangle within 
a circle. Fig. 90. Draw a diam. 1-2. From either 
end, 2, draw chords at 30° with 1-2 by Art. 35(c). 
Draw 3-4 to complete the A. 

P. M. Draw diam. 1-2. Draw sides 2-3 and 2-4 
at 30° with 1-2 by Art. 8(d) and join 3 and 4. 

43. To circumscribe an equilateral triangle about a circle. Fig. 91. Draw 
a diam. 1-2. From 2, with a rad. equal to that of the given O, cut the diam. 
extended at 3, and the given O at 4 and 5. Draw 3-4 and 3-5. With C as 
center, rad. C-3, cut 3-4 and 3-5 extended at 6 and 7. Draw 7-6 to complete 
the A. 

P. M. Draw 7-6 tangent to the O by eye; through center C draw 7-4 
and 5-6 at 30° with 7-6, Art. 8(d) . By same Art. draw 7-3 and 3-6 at 60° with 7-6. 

44. To construct a parallelogram when two sides, A-B and A-C, and the 
included angle, CAB, are given. Fig. 92. Make the sides A-B and A-C and 
the included Z CAB equal to the given sides and Z. Draw C-1 || and equal to 
A-B by Art. 36(b) and draw B-1 to complete the parallelogram. 

P. M. Make Z CAB equal to the given Z by Art. 29. Make A-B and 
A-C equal to the given sides. Draw the || sides by Art. 8(c). 

45. To construct a square on a given side, A-B. No figure. Draw a second 
side A-C J. to A-B, Art. 34(c). Draw C-1 || to A-B, and B-1 1| to A-C, by Art. 
36(b), to complete the square. 

P. M. Draw A-C and B-1 at 90° with A-B, and A-1 at 45°, by Art. 8(d). 
Draw C-1 || to A-B by Art. 8(c). 




38 



ESSENTIALS OF MECHANICAL DRAFTING 



46. To inscribe a square within a circle. Fig. 93. Draw a diam. 1-2. 
From 1 and 2 draw chords at 45° with 1-2 (Art. 35(a)) to form the square. 

P. M. Draw ± diams. 1-2 and 3-4. At 45° with these, draw sides 1-4, 3-2, 
etc., Art. 8(d). 



47. To construct a square on a given diagonal, 1-2. 
is evident from Art. 46. 



Fig. 93. The method 



Fig. 93 




8 


^ 


, 


7 


r 




/2^ 


2 




\ / / / 
Y / / 
/\ '' / 

^ — ^~- 


V 





Fig. 94 




Fig. 95 




Fig. 96 



48. To circumscribe a square about a circle. 

Fig. 94. Draw J. diams. 1-2 and 3-4. Through 1, 2, 
3, and 4 draw ||s to these diams. (see Art. 36(a) ), form- 
ing the required square. 

P. M. Draw || tangents 8-7 and 5-6, Art. 8(c). 
Through the center and at 45° with these draw 5-7, 
Art. 8(d). By same Art. draw tangents J_ to the first 
two. 

4Q. To inscribe a regular pentagon within a circle. 

Fig. 95. Draw a diam. 1-2 and a ± rad. 3-4. Bisect 
3-2 at 5. From 5, rad. 5-4, cut the diam. 1-2 at 6. 
Tlae chord of 4-6 is equal to one side of the required 
pentagon. Hence, with 4 as center, rad. 4-6, cut the 
circumference at 7 and 8. With 7 and 8 as centers, 
cut it again at 9 and 10. Draw chords 7-4, 4-8, etc., 
to form the pentagon. 

P. M. Divide the circumference into five equal 
parts by trial (Art. 15 (a)), and connect the pts. 

50. To construct a regular hexagon on a given 
side, A-B. Fig. 96. With A and B as centers, rad. 
A-B, draw arcs intersecting at 1. With 1 as center, 
same rad., describe a O cutting the first arcs at 2 
and 3. With 2 and 3 as centers, same rad., cut the 
circumference at 4 and 5. Draw chords 2-4, B-3, etc., 
to complete the hexagon. See Art. 35, Note 2. 

P. M. Draw A-5 and4-B at 60° with A-B, Art. 8(d). 
Through their intersection, 1, draw 2-3 parallel to A-B, 
Art. 8(c). Draw B-3, 2-A, 2-4, and 5-3 at 60° with 
2-3, and 4-5 parallel to 2-3. 

51. To inscribe a regular hexagon within a circle. 
Fig. 97. Draw a diam. 1-2. Draw chords 1-3, 5-2, 
6-2, and 1-4 at 60° with 1-2, Art. 35(b). Draw 3-6 
and 4-5 to complete the hexagon. 

P. M. Draw diam. 1-2. Draw diam. 4-6 and sides 
1-3, 6-2, 6-2, and 1-4 at 60° with 1-2, Art. 8(d). Draw 
sides 3-6 and 4-5 || to 1-2, Art. 8(c). 

52. To construct a regular hexagon on a given 
long diagonal, 1-2. The method is evident from Art. 51. 



GEOMETRIC CONSTRUCTION 



39 



53. To inscribe a regular octagon within a circle. 

rig. 98. Draw ± diams. 1-2 and 3-4. Bisect the 90° 
Zs by Art. 28. Draw chords 1-5, 5-4, etc., to form 
the octagon. 

P. M. Draw JL diams. 1-2 and 3-4, and diams. 5-6 
and 7-8 at 45° with these, Art. 8(d). Draw sides 1-5, 
5-4, etc. 

54. To circumscribe a regular octagon about a 
circle. Fig. 99. Circumscribe a square about the O , 
Art. 48. With pts. 1, 2, 3, and 4 as centers, rad. 1-C, cut 
the sides at S, T, U, V, etc. Join S, X, T, W, etc., to 
complete the octagon. 

P. M. Draw || tangents 4-3 and 1-2. J_ to these 
draw 1-2 and 2-3. At 45° with these draw S-X, T-W, 
Y-V, and Z-U. 

55. To construct any regular polygon. General 
Methods. 



4 


x,> 


\ Y 




3 




\/\\ 


7\ 


J 




J. 


f-^ 


/€ 


^ 


v_ 




[ ^i 


\^ 


y 




Z 


X / 


\ 


/ 


w 




i\L 


Y 


A 




1 


u 


T 




2 



Fig. 99 





Fig. 97 



Fig. 100 




Fig. 101 



(a) When a side, A-B, is given. Fig. 100. With A or B as center, rad. 
A-B, describe a semicircle upon A-B extended. Divide this arc by trial (Art. 
15(a)) into as many equal parts as the required polygon has sides (say 7). 
Through the center, B, and the second pt. of division, 2, draw B-2, which 
will be a second side of the polygon. Describe a O through A, B, and 2. (See 
Art. 40, Notes 1 and 2.) Apply A-B as a chord from 2, U, V, etc., and draw 
2-U, U-V, etc., to complete the polygon. 

Note 1. — Vertices U, V, W, and X may also be determined by radials from B, through 3, 4, 5, 
and 6, as shown. 

Note 2. — Line B-1 would be the second side of a polygon of twice as many sides. 

(b) When the ciecumscribing circle is given. Fig. 101. Draw a diam. 
1-2, and a tangent 3-4 X to it. With 1 as center and any rad., preferably 1-C, 
describe a semicircle upon 3-4. Divide this arc and draw radials as in (a). 
Join pts. U, V, W, etc., thus found, to complete the polygon. 

.P. M. Divide the circumference by trial (Art. 15 (a)), and join the pts. I, U, 
V, W, etc. 



40 



ESSENTIALS OF MECHANICAL DRAFTING 



\^\ 


■^-^//t 


Vw"''**^ 




8 


m 


3 ^ 


M ]--:• 



FrG 101 





(c) When the inscribed circle is given. Fig. 
101. Divide the circumference as in (b). Through 
these pts. of division draw radials C-W, C-X, etc., and 
obtain one side of the inscribed polygon, as X-W. 
Draw a tangent S-T || to X-W, which will be one side 
of the required polygon. The method of obtaining 
the others is evident. 

P. M. Divide the circumference as in P., M. of (b); 
then proceed as above. 

56. To construct a polygon similar to a givea 
polygon, ABCDE, upon a given side, A'-B'. Fig. 102. 

(a) When A'-B' is equal to the corresponding 
side, A-B. Draw A'-B' equal to A-B. Locate vertices 
C, D', and E' by drawing intersecting arcs with cen- 
ters A' and B' and radii equal to the distances of C, D, 
and E from A and from B respectively. 

Second Method. Divide the given polygon into 
As by diagonals A-C and A-D. Construct Zs at A' 
and B' equal to those at A and B, and at C and D', 
equal to those at C and D (Art. 29) ; then the cor- 
responding As of the given and required figures will be 
similar and in this case equal. 

P. M. Determine the vertices by Art. 57 and join 
them. 

(b) When A'-B' is greater or less than A-B. 
Draw A'-B' equal to the given length and proceed as 
in (a), second method. Then the corresponding sides 
of the given and required figures will be proportional. 

Note 1. — It is sometimes convenient to lay off A'-B' upon A-B 
or A-B extended, and then draw | |s to the other sides, as indicated 
by the dotted lines. 

Note 2. — When the ratio of A'-B' to A-B is given (as say 3 to 4) 
the length of A'-B' may be determined by a scale of proportional 
lengths. Fig. (a). Draw indefinite lines A-X and A-Y at any Z 
with each other. On one, as A-Y, set off any length, A-1, and on 
the other A-2, in this case equal to f of A-1. On A-Y set off the 
length of A-B. Draw 2-1, and B'-B || to it; then A-B' will be 
equal to f of A-B and thus be the required length of A'-B'. See 
Art. 30, Note. 



57. To plot a figure similar to a 
means of a base line or center line, 

this case an irregular curve A-F). 
any convenient position with respect 



given figure by 

and offsets (in 

Fig. 103. In 

to the given 



Fig. 103 



figure draw base line X-Y. From A, F, and any num- 
ber of intermediate pts., draw offsets J_ to X-Y. 
Having thus referred a sufficient number of limiting 



GEOMETRIC CONSTRUCTION 



41 



pts., draw X'-Y' as the line of reference for the re- 
quired figure. Upon X'-Y' set off l'-2', l'-3', etc., 
equal to 1-2, 1-3, etc., and draw offsets I'-A', 2'-B', 
3'-C', etc., equal to 1-A, 2-B, 3-C, etc. Through pts. 
A', B', C, etc., thus determined, draw the required 
curve. (See Art. lS(a).) Note application of princi- 
ple in Fig. (a), also in Fig. 132(a), (b). 

(a) To DRAW THE FIGURE TO AN ENLARGED OR 

REDUCED SCALE (say enlarged in ratio 3 to 2). The 
co-ordinate distances would be 1 = 1^ times those of the 
corresponding pts. of the given figure, and may be 
obtained by Art. 56(b), Note 2; or proportional dividers 
may be used. 




58. To draw a tangent, A-B, to a circle, through a 
given point A. 

(a) When A is on the circumference. Fig. 
104(a). Draw rad. C-A, and tangent A-B J_ to it, 
Art. 34(c). 

P. M. Draw A-B J. to rad. C-A by Art. 8(d). 

(b) When A is on the circumference and the 
center inaccessible. Fig. 104(b). With A as center, 
any rad., cut the curve at 1 and 2. Draw chord 1-2. 
Draw A-B || to 1-2 by Art. 36(b). 

P. M. Obtain chord 1-2 as above. Draw A-B |1 
to it by Art. 8(c). 

(c) When A is outside of the circumference. 
Fig. 105. Join A to center C. Upon A-C draw a 
semicircle to determine pt. of tangency B. Draw tan- 
gent A-B. See Art. 34(c), Note 2. 

Note. — A second tangent, A-B', may be drawn, as indicated. 

P. M. Draw A-B tangent to the O by eye. Locate 
pt. of tangency by a rad. C-B ± to A-B, Art. 8(d). 

59. To draw a tangent, A-B, to two given circles. 
Fig. 106. Draw line of centers 1-2. Subtract length of 
rad. of smaller O from that of the larger and draw 
concentric arc 3-4. Obtain tangent 2-3 by Art. 58(c). 
Draw rad. 1-A through 3, and make Z B 2 5 equal 
to Z A 14, Art. 29. Draw A-B, the required tangent. 

Note. — To draw a tangent, C-D, passing between the centers — 
add length of rad. of smaller O to that of the larger, and draw con- 
centric arc 6-7. Draw tangent 2-7, and rad. 1-7, locating pt. C. 
Make / D 2 1 equal to Z 2 1 7, thus locating D. Draw C-D. 

P. M. Draw tangents A-B and C-D by eye. Locate 
pts. of tangencj' as in P. M. of Art. 58(c). 




Fig. 105 




42 



ESSENTIALS OF MECHANICAL DRAFTING 



60. To draw a circular curve of given radius, D-E, tangent to a given circular 
curve and to a given straight line, A-B. Fig. 107. At distance D-E from A-B 
draw a || 1-2. Upon any rad., C-3 extended, set off 3-4 equal to D-E. Con- 
centric with given curve, and with rad. C-4, draw an arc cutting 1-2 at 5, the 
center of the required curve. Line C-5 and a J_ to A-B from 5 determine pts. of 
tangencj' 7 and 6. See application in Fig. (b) at G. 

Note. — The method of drawing a circular curve of given rad. tangent to two given circular curves 
is evident from Fig. (a) . 

61. To draw a circular curve tangent to two given straight lines, A-B and 
C-D. Fig. 108. 

(a) When a point of tangency, E, on A-B is given. Fig. (a). Extend 
A-B and C-D to intersect at 1, and make 1-E' equal to 1-E. At E and E' 
draw ±s to intersect at 3, the center of the required curve; or bisect the included 
Z BID and draw _L at E to cut the bisector 1-2 at 3. 

Note 1. — When the vertex, 1, is inaccessible, the bisector may be obtained by Art. 28 Note. 
Note 2. — The construction, when A-B and C-D are ||, or at right Zs, is evident from Figs, (b) 
and (c). 




Fig. 107 




(C) M 

Fig 108 



(b) When the radius, 3-E, is given. Fig. (a). 
At distance 3-E draw ||s to A-B and C-D, intersecting 
at center 3 ; or obtain bisector 1-2 as in (a) and a 1 1 to 
one side intersecting 1-2 at 3. See application in Fig. 
107(b) at F. 

Note. — ^WTien A-B and C-D are at right Zs, the rad. may be ap- 
plied as in Fig. (c). 




Fig no 



62. To draw a circular curve tangent to three straight lines, A-B, A-C, 

and C-D. Fig. r09. The construction is identical with that of Art. 42. 



GEOMETRIC CONSTRUCTION 



43 




63. To draw circular curves tangent to two given parallel straight lines, 
A-B and C-D, at B and C, and to each other at any point E, in line B-C. Fig. 110. 
Draw ± bisectors of B-E and E-C. At B and C draw _Ls to A-B and C-D to 
cut the bisectors at 1 and 2, the centers of the required arcs. 

64. To draw an ellipse when the axes, A-B and C-D, are given. 

(a) By Focal Radii. Fig. 111. Draw the axes _L to each other at their 
middle pts. With either end, C, of the minor axis as center and rad. equal to 
^ of the major axis, cut the latter at F and F', the foci of the elhpse. Between 
the center, E, and either focus place any pt., 1. With the foci as centers and 
rad. A-1, draw arcs upon opposite sides of A-B. With same centers and rad. 
B-1, cut these arcs at 2, 3, 4, and 5, which will be pts. of the required curve. Assume 
a sufficient number of other pts. for focal radii 
on A-B and proceed in like manner. Draw the 
curve through the pts. thus found, freehand. 
See Art. 18(a). 

Note. — The method of obtaining a tangent, T-T', at a 
given pt., 2, in the curve is e\'ident. See Art. 2(d). 

(b) By Trammel Method. Fig. 111. On 
the St. edge of a piece of paper mark off G-I 
equal to | of the major axis, and I-H equal 
to f of the minor. Moving this trammel so 
that pt. G remains on the minor axis and H on 
the major, set off a sufficient number of pts. for 
the successive positions of I to determine the 
required curve. 

Note. — Having located pts. for J of the cur^'e, corre- 
sponding pts. could be determined by Art. 57. 

(c) By Revolution OF A Circle. Fig. 112. 
Upon the axes describe O s. Divide these O s 
into the same proportional parts by diams. 
If the large O be imagined to revolve about 
axis A-B, pts. 1, 2, etc., will appear to movej. 
to A-B. When 1 and 6 coincide with D and 
C, pts. 2, 3, etc., will have moved proportional 
distances which are determined by ||s to A-B 
from the corresponding pts. 2', 3', etc., of the 
smaller © , at 2", 3", etc., of the required curve. 

Note.— The figure indicates a second method of obtain- 
ing a tangent at a given pt. in the curve. 

(d) By Parallelogram Method. Fig. 113. 
Draw lis to A-B and C-D through A, B, C, 
and D, forming a parallelogram. Divide A-F 
into any number of equal parts and A-E into 
the same number of equal parts. Through 
the pts. of division on A-F, draw D-1, D-2, D-3. Through the corresponding 
pts. on A-E draw C-1, C-2, C-3 intersecting the lines from D at 1', 2', 3', 





Fig 

1 


. Ill 


^ 




2/\ 


.xJ 


\ 




//^ 3' 


^ 


^4 \^/ 


^^^^■/ .4" 


Jb 


\/ 


x^ 


/ 


Fig 112' 


6 


V 






Fig. 113 



44 



ESSENTIALS OF MECHANICAL DRAFTING 



which will be pts. in the required ellipse. In 
like manner find pts. for remainder of curve. 

Note 1.^ — The same construction appKes when any two 
conjugate diameters are given. 

Note 2. — The method of inscribing an elUpse in any- 
given parallelogram (not square) is evident. 

(e) By Circular Arcs. Fig. 114. The 
following is one of several methods of approx- 
imating an ellipse: Draw D-B. Make 1-2 
equal to 1-D and D-3 equal to 2-B. Draw 
a _L bisector to 3-B, cutting C-D extended at 
4, and A-B at 5. Make 1-5' equal 1-5 and 1-4' equal 1-4, and draw 4-6', 4'-6", 
and 4-6'". With 5 and 5' as centers, rad. 5-B, draw arcs 6-6'" and 6"-6'; then 
with 4 and 4' as centers, rad. 4-D, draw arcs 6'-6 and 6"'-6" to complete the 
curve. 




Fig 114- 



CHAPTER V 



ORTHOGRAPHIC PROJECTION 



65. General Principles. It has been noted that mechanical drawings are 
made for the purpose of showing the exact facts of form, dimension, and arrange- 
ment of parts in objects of a structural character. (See Art. 1.) To express these 
facts fully and clearly it is necessary to represent the object by two or more 
related drawings each of which gives certain information that the others lack. 

These drawings, though made upon one plane, the paper, by the methods of 
Chap. IV, are regarded as projections of the image of the object upon different 
planes || to the axes or principal dimensions of the object, and imagined to be 
obtained by means of ±s to those planes from all pts. of the object; that is, in 
accordance with the principles of Orthographic (true drawing) Projection. 

(a) In Fig. 115(a), ABCD is a pictorial drawing of a hollow rectangular 
block, placed squarely in front, with the center of the opening on the level of the 
eye. By experiment with a similar object, it will be observed that the front sur- 
face, being at right Z s to the direction in which it is seen, appears in its exact form. 

The surfaces of the opening, although at right Z s with the front, appear fore- 
shortened and to incline towards each other; the farther lines appear shorter than 
the nearer, and the receding lines to incline and converge towards a pt. at the center. 

The apparent decrease in size, 
foreshortening, inclination, and con- 
vergence is due to the position of the 
Unes and surfaces of the object with 
respect to the eye of the observer, and, 
obviously, if the position of the ej'e be 
changed, the appearance will also 
change. Thus, if the eye be moved 
a certain distance upward, the object 
would appear as shown in Fig. (b) 
and if moved upward to the right, as 
in (c). 

It is evident that none of these 
pictorial representations shows the 
exact form, size, and relation of all 
the hues and surfaces. 

(b) Let LMNO (Fig. 115(a)) 
represent a vert, pane of glass or 
wire screen placed |1 to the front 
surface of the object. Consider 
this glass or screen to be merely piG. I 15 








(b) 



(c) 



46 



ESSENTIALS OF MECHANICAL DRAFTING 



a plane, and the picture ABCD, as obtained by tracing lines upon the 
plane to exactly cover the outlines of the object seen through it. Since the plane 
is nearer to the eye than the object, and the rays of light converge from the 
object to the eye, it follows that all lines of the tracing must be shorter than those 
of the object; also, that if the object be projected forward until its front surface 
coincides with the plane, the tracing of that surface would be identical both in 
form and dimensions with the surface itself. 

(c) If now, instead of moving the object, lines be imagined to extend from 
all its pts., A, B, C, D, etc., _L to the plane, as shown pictorially in Fig. 116, these 
±s would intersect the plane at A^, B^, C^, D^, etc. These pts. are called 
projections of the pts. of the object, and lines A^-B^, B^-C^, etc., joining them, will 
be projections of the lines and surfaces of the object. 

The plane is a plane of projection and the J_ s are the projectors of the pts. 



















N 


D* 


, WIDTM _-j 
OR BREADTH 


F» 














kJ 

1 














L 


A" 


H» 




BVG* 




M 



Fig. 116 



riG.117 



(d) Since the projectors are || to each other, and the rear pts. of the object 
are perpendicularly back of the front pts., it follows that the projections of the 
front and rear pts. will coincide, as indicated in the figure. The projection, when 
viewed squarely, as in Fig. 117, thus shows what the eye would see if imagined to 
be directly opposite each pt. of the object at the same time; namely, the exact 
form and dimensions of the front and rear surfaces, and the dimensions of the 
object from left to right and bottom to top. 

(e) Although in this case the first dimension is the length and the latter the 
width, it is convenient in speaking of the dimensions of an object in a definite 
position to call the hor. dimension from left to right the width, or breadth, that 
from front to back the depth, and the vert, dimension the height, regardless of 
their extent. 

In the notation of this proj . and others which will be explained, the first letter 
in each instance indicates the pt. of the object nearer the plane, and the small 
letter the plane upon which the proj. of that pt. lies. To avoid confusion, the 
notation of the opening is omitted. 

(f) Since the three dimensions of an object are _L to each other and a plane 
has but two dimensions, it follows that when two dimensions of the object are 



ORTHOGRAPHIC PROJECTION 



47 



projected in their exact size the remaining dimension is not seen; that is, no more 
than two dimensions can be shown in their exact size and relation in one proj. 

Hence, to show the dimension from front to back (depth), another proj. upon 
a plane _L to the front or vertical plane must in like manner be obtained. 

Thus in the pictorial illustration Fig. 118, D'^C'^F^E", etc., is the proj. of 



the object upon a to-p or horizontal -plane ONQP, and B G F 
upon a side vertical or profile plane 
MRQN, at right Z s to the first two. 
Note. — The relation of the planes may be 
illustrated by means of a paper box, hinged 
panes of glass, or screens. 

(g) To show these projs. as they 
are generally arranged in a mechan- 
ical drawing, the planes of two of the 
projs. are imagined to be revolved 
about their hnes of intersection into 
the plane of the other regarded as the 
plane of the drawing, as in Fig. 119. 

(h) It should be noted that the 
projs. upon the top and side planes 
are precisely the same as would be 
obtained upon the front plane if the 



etc., its proj. 




FiG.118 



p 

E" 


Q 
H" F"G" 


\ 


\ FiS.IIQ 

\ 
\ 
\ 

Q.'l 


D" 









t 

r 
1- 
a 

o 
1 


"^\ 


A" 


G.L. 


B" 3 

u 

N 


\ 

\ 

HTofP 


Dv 


Ev 


1 

WIDTH ^ 
OR BREADTH 

1 

1 ''" 


V 

Fv 


% P 

1- 
> 

2 CP 


3 

-DEPTH- 
DP' FP 


EP 




1 
1 


t 










l- 

r 
o 

UJ 

r 








r 





6*0" 



M 



BPAP GPHP 



48 ESSENTIALS OF MECHANICAL DRAFTING 

object were turned from the position given so that its dimensions of width and 
depth, and then its height and depth, are 1 1 to that plane. 

The opening cannot be seen when the object is looked at squarely, either from 
above or from the side, but as it is necessary to show its projs. upon the top and 
side planes, — the lines whose projs. do not coincide with those of visible lines 
are represented by dashed lines, to indicate that they are invisible or hidden. 
Observe that the outer lines of all projs. represent visible lines of the object;' those 
within represent hidden lines when seen in the other projs. to lie below or behind 
some solid portion. 

(i) It is evident that the three dimensions are shown by any two of the 
projs.; that two ± projs. are, therefore, necessary to show the three dimensions; 
and that the three projs. together determine completely and clearly the exact 
form, size, and relation of all lines and surfaces of the object. 

(j) These three mutually _L planes of proj. are called the co-ordinate planes, 
and for brevity are denoted by V, H, and P respectively. 

A proj . is named from the plane upon which it is imagined to be obtained, not 
from the particular part seen or shown. Thus those upon V, H, and P are respec- 
tively the vertical, horizontal, and profile projection. 

In practical drafting, the first is called a front elevation ov front vieiv, the second 
a plan or top view and the third a side elevation or side view. 

The terms "plan" and "elevation" are used chiefly in architectural drafting; 
the term "view" is more generally used. 

(k) The line of intersection of V and H is called the ground line or trace of 
V and H, and is denoted by G L. 

When looking squarely at V, the G L represents H; and when looking at H, it 
represents V. The line of intersection of V and P is called the vertical trace of P; 
that of H and P is the horizontal trace of P. These are denoted bj^ V T of P and 
H T of P. 

When looking at V or H, these traces indicate the position of P. When looking 
at P, they represent V and H. The verts, and hors. from the views to the traces 
represent the projectors to the planes. 

(1) When H and P are revolved, the front and top views of any pt. are seen 
to lie in the same vert.; the front and side views in the same hor.; and its top and 
side views equally distant from the G L, and V T of P, respectively. 

Rules 1, 4, 5, and 7 (Art. 74) may here be noted. 

(m) Views could in like manner be obtained upon planes auxiliary to the 
co-ordinate or principal planes, V, H, and P; namely, upon a profile plane at the 
left of the object, upon a bottom plane || to H, upon a rear plane || to V, and 
upon planes oblique to either two of the co-ordinate planes. 

We may thus have front, top, right and left side, bottom, rear, and oblique 
views. 

With the exception of the latter, the relative arrangement of these views in 
a mechanical drawing would generally be as indicated in Fig. 120. See Art. (i). 
Note that the line of each view (excepting the rear) nearest the front view 
represents the front line or surface of the object in that view. 



ORTHOGRAPHIC PROJECTION 



49 



66. To draw the front, top, and side views of a rectangular object. Suppose 
it is required to draw tiiese views of the block (Fig. 119(a)), its position and dimen- 
sions being given. 

(a) First draw ONQ' and MNQ to represent the traces of V, H, and P. 
That portion of the paper below 0-N and to the left of M-N will represent the 
front plane, that above 0-N and to left of N-Q the top plane, and that to the 
right of M-N and below N-Q' the right side plane. The outer limiting lines of 
the planes are always omitted. 



E" 


H« 




F" 


G" 






Q 

— 


u. 
O 


^^riG.ii9( 




' i 
1 
1 1 
1 1 
1 1 




UJ 
Q 

♦ 


D" 

o 


A" 


GL. 


C"|B" 

1 1 


h 


H 
/ 


IF" 
N 


htWp. 1 




E" 


WIDTH 
OR BREADTH 


C 


F' 


\ 






p 


[-DEPTM»| 




D" 




D' f 


E" 














a 












1 E 

o 

r 


> 










j__ 





Q' 



^c," 



Ml 



B'-A" 



CK' 



(b) Since the distance of the object from the planes is immaterial, the views 
may be drawn anj- distance from the G L and traces of P. Hence, at any con- 
venient distance below 0-N and to the left of M-N, draw rectangle A^B^C^D^ 
for the main lines of the front view, making the hor. and vert, dimensions 
equal to the breadth and height respectively, of the object. The opening should 
at first be disregarded. 

(c) The top view of each pt. must be in a vert, through its corresponding 
front view. Art. 65(1). As in this case all pts. of the front face are equally distant 
from V, di'aw the projectors from the front view, and at any convenient distance 
above the G L, draw the hor. D.^-C^ for its top view. 

As the rear face is || to the front, its top view will be E^-F^, at a distance 
from D^-C^ equal to the depth of the object. As the side faces are _L to H, their 
top views will be the verts. D^-E^ and C^-F^, thus completing the rectangle 
D^C^F^E^ for the main lines of the top view of the object. 

(d) From Art. 65(1) it follows that the side view of each pt. will be in the 
hor. through its corresponding front view and at the same distance from the V T 
as its top view is above the G L. Hence, draw indefinite hors. from the front 
view, and from the top view to N-Q'. 

Then as N-Q and N-Q' both represent the H T of P, describe arcs with N as 
center, to carry the pts. from N-Q to N-Q'. From these pts. draw verts, to 
intersect the projectors. from the front view, thus determining B^, G^, F^, and 
C and forming the main lines of the side view. 



50 



ESSENTIALS OF MECHANICAL DRAFTING 



(e) Similarly, if the side view is first obtained, the reverse of the above process 
will determine the top view. 

(f) Now, returning to the front view, draw the rectangle to represent the 
opening, then project as before, to complete the other views. 







TOP VIEW 


























LEFT SIDE VIEW 




FRONT VIEW 




RIGHT SIDE VIEW 


REAR VIEW 




















































































BOTTOM VIEW 














Fig. 120 
















(a) 






Cb) 




Fig. 121 

(g) Practical Methods. In practical drafting, only such portions of the 
projectors are penciled as may be necessary to locate the required pts. The 
traces of the planes of proj. are also omitted; center, base, or other hnes || to 
the positions of the traces being utilized as reference Unas in locating the views, 
practically as though such lines were the actual traces of the planes. Thus, in 
determining the views (Fig. 119(a)) the lower hor. of the top view and left vert, of 
the side view could have been located any convenient distance from the front 
view, and the depth set off from top to side or vice versa, by means of the dividers. 

Fig. 121(a) shows the views with the traces and projectors omitted. In 
this case aU measurements were set off with reference to lines representing the 
axes of the object, called center lines. 



ORTHOGRAPHIC PROJECTION 



51 



It is evident that a C. L. of a view may be regarded as the trace of a central 
plane ± to the plane of that view. 

Fig. 121(b) shows the same object in a different position. Working drawings 
of an object composed of several rectangular parts are shown in Figs. 179, 180. 

67. Objects Having Surfaces Oblique to the Co-ordinate Planes. The form 

of an object is frequently such that some of its surfaces and lines will be oblique 
to the planes of one or more of the views; and, therefore, foreshortened in those 
views. Thus in the pyramid (Fig. 122) the front and rear faces are oblique to V 
and H, the left and right faces to H and P, and its slant edges to V, H, and P. 
These surfaces and lines are, therefore, foreshortened in all of the views. 

The base is 1 1 to H and has two of its edges 1 1 to V and two 1 1 to P. When thus 
placed the object is || to V, H, and P as much as it can be, since its axes or principal 
dimensions are also || to those planes, and the views enable the form of the object 
as a whole to be determined, as in the case of a rectangular object. 

Rules 2, 3 and 6 (Art. 74) may here be noted. 




- Fig. 122 



BC BA 




68. Objects xiaving Curved Surfaces. 

(a) Any view of a sphere is a O equal to a great O of the sphere. Fig. 123. 
Observe that the great O ABCD is || to V, AECF || to H, and DEBF l| to P. 

The view of a right circular cjdinder upon a plane to which its axis is± is a ©. 
Its view upon a plane || to the a.xis is a rectangle. Thus in Fig. 124 the bases 
are || to H and X to V; thej' are, therefore, seen upon H in their exact shape, 
and upon V as || hors. equal to the diam. of the base. The elements are || to V 
and _L to H and, therefore, seen upon V in their exact length, and upon H as pts. 
in the O . 

The views of a right circular cone whose axis is thus related to the planes are 
aO and a A. Fig. 125. The outer elements only being || to V, they only are 
seen upon it in their exact length. Intermediate elements are unequally inclined 
to V and, therefore, unequal upon it. As all the elements are equally inclined 
to H, they are equally foreshortened upon H. Note that as the third view in each 
case would be the same as one of the others, its representation is unnecessary. 



52 



ESSENTIALS OF MECHANICAL DRAFTING 




(b) Equidistant elements of a right 
circular cylinder or cone may be deter- 
mined by equal division of the base or 
other O of the surface, in the view in 
which that O is seen in its exact shape. 
Twelve are usually sufficient. 

(c) Fig. 126 represents a truncated, 
cylinder. As the elements are _L to H, 
the top view of the curve of the oblique 
surface coincides with that of the base. 
As the surface is oblique to P, its side 
view will also be a curve. To draw 
this view it is necessary, since there are 
no vertices as in rectilinear figures, tO' 
obtain a sufficient number of its deter- 
mining pts., by assuming these first in 
the known (front and top) views of 

the curve. Having located these pts. in the side view the curve is drawn 
through them, as shown. See Art. 18(a). 

In cylindric or conic surfaces these pts. maj' be regarded as the ends of elements. 

69. Projections upon Oblique Auxiliary Planes. When it is necessaiy to 
represent an object so that some particular surface, ± to one plane but oblique 
to another, shall be shown in its true or exact shape, an auxiliary view upon a plane 
II to that surface may be obtained. 

(a) In Fig. 127, D^B^C'^, etc., 'represents a proj. upon an aux. plane A, 
II to the surface DBC, that is, upon a plane _L to V, but oblique to H. The 
method of obtaining this view differs from that of obtaining a side view only in 
that the V T in this case is 1| to D^B'^'C^, instead of ± to the G L. Having 

located the traces of A, 
draw ± s to the V T from 
the front view and inter- 
sect these by projecting 
from the top view as 
shown, or transfer the pts. 
with dividers as in Art. 
68(g). Joining the pts. 
thus determined will form 
the required view. Note 
application of principle in 
Figs. 144, 182. 

(b) In Fig. 128(a) an 
aux. view A^B^D^, upon a 
plane, A, ± to H and || to 
a central plane, Y-Y, is 
shown. In this case the 
view was determined by 




Fig 127 



ORTHOGRAPHIC PROJECTION 



53 



means of C. Ls. and base lines regarded as the traces of planes J_ to those of the 
views. The method differs from that of obtaining the front view in Fig. 127 only 
in the changed positions of the central planes X-X and Y-Y. Thus Y-Y is the 
H T of the given central plane; X-X is the H T, and trace upon A, of a second 
central plane _L to Y-Y. 
Z-Z is the V T of the 
plane of the base, and 
Z'-Z' drawn anj^ con- 
venient distance from 
Y-Y and || to it is the 
trace of the base plane 
upon A. 

Perpendiculars to 
Z'-Z' from A^, B", and 
C^ determine A^, B^, 
and €"*■; setting off the 
height of D above Z-Z 
fi'om the front view and 
joining D*, A-*-, and B^ 
completes the required 
view. 

(c) Fig. 128(a) also 
method of obtaining an 
jQB^BgB^ upon a plane, B, 




fiG 128 



shows the 
aux. view 
II to a sur- 
face DBC, which is not ± to the plane 
of either of the given views, that is, not 
shown as a line, but ± to which a 
plane, Y-Y, can be directly determined. 

The aux. view A^B'^D^, is first ob- 
tained as in the preceding case. Y'-Y' 
II to D'^B^C^ is the trace upon B of the 
± plane Y-Y. 

Perpendiculars to Y'-Y' from A^ 
and D^ determine A^ and D^; setting 
off the distances of B and C from Y-Y, 
± to Y'-Y', and joining pts. A^, B^, 
C^, D^ completes the required view. 

(d) Fig. (b) shows the method of 
obtaining an aux. view upon a plane, 
B, A. to the axis of an object which is 
oblique to V, H, and P. Note that the 
views upon A and B determine the true dimensions of the prism. 

(e) In a curviUnear object, it is necessary to assume pts. in the known views 
of the curves and then obtain these in the aux. view, as is evident from Art. 68(c). 

(f) When the true shape of a particular surface only is required the same 
method, or that of Art. 70, maj^ be used. 




54 



ESSENTIALS OF MECHANICAL DRAFTING 




70. Revolution of Surfaces. It 

is frequently desirable to determine 
the true shape of a surface by revolv- 
ing it about an axis until 1 1 to V, H, 
or P. 

(a) Fig. 129(a) represents a 
truncated hexagonal prism with its 
oblique surface ABCDEF revolved 
about an imaginary axis _L to V 
through pt. A, until it is || to H, upon 
which it is, therefore, seen in its true 
shape A'B'C'D'E'F'. 

In revolving a surface, each pt. not 
in the axis of revolution describes an 
arc whose plane is _L to that axis. 
In this case, therefore, the arcs are 




r IG. 129 H' G" H» 



seen as arcs in the front view and as lines || to the G L, or C. L., A^-D', in the 
top view. Hence, the front view of the surface remains a line of the same length, 
and the distances of the pts. of the surface from V also remain unchanged. 

The surface could in like manner be revolved 1 1 to H, or to P, about any other 
axis ± to V. Thus A"B"C"D"E"F" represents it revolved about 1-2 until 1 1 to P. 

Fig. (b) represents the surface revolved about its side B-C until || to V. 
As B-C is II to V, the front views of the arcs described by A, F, E, and D are _Ls 
to B-C at A^, B^, C^, and D^; and as the true distances of A, F, E, and D back 
of B-C are seen in the top view, these distances are set off on the ±s at A', F', 
E', and D' as shown. 

The figure also shows a lateral surface revolved about its base edge, G-H, 
until II to H; and, about the lateral edge, A-G, until || to V. Note application of 
principle in Fig. 140 to the revolution of an imaginary surface on C-D. See also 
Figs. 145, 153, 158, 159, 161(b). 



ORTHOGRAPHIC PROJECTION 



55 



(b) In Fig. 128(a), a surface DEC, oblique to V, H, and P, is shown revolved 
II to H about its base edge B-C. As B-C is jj to H, the top view of the arc 
described by pt. D will be a i. to B^-C^, coinciding with Y-Y. Hence, the 
true distance of D from B-C is determined, in this case, by an aux. view upon 
a plane A, || to the central plane Y-Y, to which the surface DBC is ± . Art. 69. 
Rule 8 (Art. 74) may here be noted. 

7 1 . True Length and Position of Lines Oblique to the Co-ordinate Planes. To 

determine the true length of a Hne oblique to the planes, the hne may be projected 
upon a II aux. plane, or revolved || to a plane of proj., practically as in the case 
of a surface. 

(a) First Method. To obtain the 
true length of A-D (Fig. 128(a)) upon an 
aux. plane J_ to H. The method is evi- 
dent from Art. 69(b). A*-D^ is the true 
length, and the Z it makes with the H T 
of plane Y-Y containing the line is the true 
Z the line itself makes with H. 

To obtain the true length of A-D (Fig. 
130) upon an aux. plane A _L to V, regard 
A^ -D^ as the V T of the plane containing 
the line; Y-Y as the H T of a central plane 
± to the first; and Y'-Y' placed any 
distance from A^-D^ and || to it, as the 
trace of this second plane upon the aux. 
plane A. 

Projecting from A^ and D^ J. to Y'-Y' 
obtain D^, and set off the ± distance 
A*-D^ will be the required true length and 
makes with V. 

(b) Second Method. If the line A-D be revolved || to V about an axis ± 
to H through D, the line remaining at the same Z with H and pt. D fixed, 
pt. A will describe a hor. arc, the top view of which will be the arc A^-A', and the 
front, the hor. A^-A'. In this position, therefore, the top view of the line remains 
unchanged in length, while the front A'-D^ shows the true length of A-D and the 
true Z (b) that it makes with H. 

As all of the edges meeting in the vertex D are equal and equally inclined to 
H, A'-D^ is the true length and (b) the true Z of all. 

Again, as all pts. in the revolved line remain on the same levels, they describe 
similar hor. arcs. The true distance from D of any pt., as E on D-C, may, there- 
fore, be found by drawing a hor. from that pt. to the true length line A'-D^, as 
proved by the top view which shows E revolved into A'-D^. E'-D^ is, there- 
fore, the true length of E-D. 

The figure also shows the line A-D revolved 1 1 to H about an axis ± to V, 
through D. A"-D^is, therefore, the true length and (c) the true Z of A-D with V. 

The line could in like manner be revolved about an axis through A. 




Fig 130 



that A is back of Y-Y. Then 
(a) the true Z that the line itself 



56 



ESSENTIALS OF MECHANICAL DRAFTING 



72. Objects Ob- 
lique to the Co-ordi- 
nate Planes. It fre- 
quently occurs that 
the axes of one or 
more integral parts of 
an object are oblique 
to those of the main 
portion, and, there- 
fore, to the planes of 
one or more of the 
views. Thus, in Fig. 
193 the left and right 
side pieces of the taboret are oblique to V and P; 
in Fig. 194 the braces (A) are oblique to V and H, 
and braces (B) to H and P (see also bolts in Fig. 
182) ; in Fig. 199 the legs are obhque to V, H, and P. 
When the detail or part is || to V, H, or P 
only, its view upon that plane should, as a rule, 
first be determined, since that view only will 
show two of its dimensions in their exact size. 

When not || to either V, H, or P, it may, if 
necessary, be represented first as || to one of those 
planes and then as revolved to the required posi- 
tion; or, the method of Art. 73 may be used. 

(a) In Fig. 131, (a) represents a rectangular 
prism II to V, H, and P. (b) represents this 
prism II to H but oblique to V and P, that is, 
turned backward at the left from the position of 
(a), through an Z of 30° about an axis _L to H. 
(c) represents it 1 1 to V but oblique to H and P, 
that is, turned to the right from the position of 
(a), 30° about an axis J_ to V. (d) represents it 
II to P but oblique to H and V, that is, turned for- 
ward from the position of (a), 30° about an axis 
J. to P. 

(b) By experiment with the actual object, it 
will be o'bserved that the positions of its lines rela- 
tive to the plane to which the axis of revolution is 
± remain unchanged. It follows that the object 

may be turned through any Z without changing the form and dimensions of the 
view upon that plane. Thus, in Fig. 131, the top view in (b), the front view in 
(c), and the side view in (d) are the same as the corresponding views in (a), 
save that they show the changed relation of the object to the planes to which 
the axis of revolution is ||. The views upon the latter planes change in outline 
with each position of the revolution, but as no pt. changes its distance from the 





ORTHOGRAPHIC PROJECTION 



57 



plane to which the axis is J. , the dimen- 
sions II to the axis also remain un- 
changed. Thus, corresponding pts. of 
the front and side views in (b) and (a) 
are at the same distances from the G L, 
and H T of P ; those of the top and side 
views in (c) and (a) at the same distances 
from the G L, and V T of P; and those 
of the front and top views in (d) and (a) 
at the same distances from the traces of 
P. Note Rule 8, Art. 74. 

(c) In determining the views of an 
object thus revolved, having located 
the G L and traces of P, or equivalent 
reference Hnes*, draw the main axis 
or axes of the unchanged view at the 
required Z s with the planes to which 
the object is oblique. (See lines A-A 
andB-BinFig. (b).) Measuring upon, 
and 1 1 to these axes, construct the view. 
From this view draw projectors to a 
second plane and set off upon them the 
unchanged dimensions" 1 1 to the axis of 
revolution, measuring from the trace of 
the first plane, or equivalent reference 
hne. Joining the pts. thus determined 
will complete the second view. From 
these, the third may be obtained as in 
Art. 66. 

(d) In Fig. 132, (a) and (b) repre- 
sent the same prism when obhque to V, 

H, and P. (a) represents it as turned to the right from the position of (b) in 
Fig. 131, 30° about an axis ± to V; (b) as turned backward from the position 
of (a), 30° about an axis J_ to P. 

In determining these views the same principles apply as in the preceding cases. 
By assuming and representing an object in a primary position, as nearly as may 
be like that reciuired, and then as revolved, any conceivable position may be 
represented. 

Observe that the side view in Fig. (b) was plotted from Fig. (a) by Art. 57. 

(e) In an object having curved surfaces, pts. must be assumed in the 
primary positions of the curves, as explained in Art. 68(c). Note applications 
of principle in Fig. 193. 




*In practical work the traces and projectors are omitted, as explained in Art. 66(g). 
shown in Fig. 131 merelj' for illustration. 



They are 



58 



ESSENTIALS OF MECHANICAL DRAFTING 



(f) Instead of representing each position separately, the required views may 
be obtained by revolution about a C. L. of a primary position as in Fig. 133(a), 
or about an axis || to that C. L. as in Fig. (b). 

73. Partial Views, and Use of Auxiliary Views in Determining Required Views. 

(a) Partial views, and aux. views (views of revolved, lines or surfaces, and 
oblique views) may frequently be used in place of complete or regular views 
and thus economize space and labor. Thus in Fig. 214 half of the top view 
would suffice to determine the required pts. of the front; in Fig. 153 a revolved 
half -base is used for the same purpose; in Fig. 147 half only of the side view" 
of A and C is necessary; in Fig. 144 half of the oblique view would suffice; in 
Fig. 145 the revolved half-bases of (B) and (C) dispensed with a side view; in 
Fig. 146 a view _L to the axis of (B) or a revolved imaginary © is necessary to 
determine positions of elements. 

(b) In representing objects oblique to the co-ordinate planes, primary po- 
sitions may be omitted and the missing dimensions of the required views deter- 
mined by the same general methods. 




FfG. 133 




To illustrate, Fig. 134(b) represents a pentagonal prism with its axis and 
lateral edges j] to H and at 30° back to the right with V. Its bottom face also 
is II to H. Instead of representing the prism first as || to V and H as in Fig. (a), 
and then as revolved about an axis _L to H until its lateral edges make the required 
Z of 30° with V, the representation of the first position may be omitted and 
the missing dimensions determined as follows: — 

First locate X-X, Y-Y, Z-Z, and Y'-Y' for the C.Ls. of the required views. 
Through the center of the top view draw V-V at 30° with Y-Y for the H T of a 
central plane containing the axis and W-W for the H T of a central plane ± to 
V-V. Equally distant from the center and at a distance apart equal to the axis 
of the prism, draw indefinite _Ls to V-V for the bases. Now obtain an aux. view 
upon a plane || to the central plane W-W. The H T of this plane would be || 



ORTHOGRAPHIC PROJECTION 



59 



to W-W, and the trace of the hor. central plane Z-Z, upon the aux. plane, would 
also be 1| to W-W. Hence, at any convenient distance from either base, draw 
this trace Z'-Z' and about its intersection with V-V construct the view, noting 
that as the lower surface of the prism is || to H, the hne A-B will be || to Z'-Z', 
as shown. Projecting perpendicularly to W-W from this view, complete the top 




view. Project next to V and compltte the front view, obtaining the distances of 
pts. above and below Z-Z by measuring from Z'-Z' in the aux. view. The missing 
dimension in this case could also be obtained by revolving a base || to H, then by 
counter revolution projecting back to the top view, as shown. 

In some cases a view upon an aux. plane || to the central plane V-V, as 
shown, would be necessary instead of, or in addition to, that of the end. 



60 



ESSENTIALS OF MECHANICAL DRAFTING 



Fig. 135 represents the same prism with its axis || to V and at 60° up to the 
right with H. Its rear face also is || to V. 

Fig. 136 represents it with its axis || to P and 30° forward with V. Its left 
face also is II to P. 




The methods of obtaining the missing dimensions are identical with those of 
the preceding case and are shown by the figures. 

(c) Fig. 137 represents the prism with its axis inclined up to right at 60° 
with H and in a plane at 45° back to left with V. One of rear faces also is at 45° 



ORTHOGRAPHIC PROJECTION 



61 




Fig. 137 



with V. This is identical with 
a revolution from the position 
of Fig. 135, 45° about a vert. 
axis, but instead of represent- 
ing the prism first at 60° with 
H, as in Fig. 135, the work in 
this and similar cases may be 
shortened as follows: — 

First locate the C.Ls. X-X, 
Y-Y, and Z-Z for the required 
views. Through the center of 
top view draw U-U at 45° for 
the H T of a central plane con- 
taining the axis, and V-V ± to 
U-U for the trace of a central 
plane ± to H and to an aux. 
plane A 1 1 to the axis. Through 
any convenient point on V-V 
draw W-W at 60° with U-U as 
the C.L. of the aux. view, and 
about the intersection of V-V 

and W-W construct the aux. view as in Fig. 135. From these aux. views obtain the 
top view. Observe that this is identical with revolving both front and tap views 
of Fig. 135, through an Z of 45°. Project next from H to V, and obtain the dis- 
tances of pts. above and below Z-Z by measuring from Z'-Z', regarded as the trace 
of the hor. central plane Z-Z upon the aux. plane. Connect the proper pts. to 
complete the front view. 

74. Rules Governing the Position of Lines and Surfaces Relative to Any Two 
Perpendicular Planes of Projection. 

(1) A line ± to a plane is 1| to the JL plane. Its view upon the first is a pt.; its view upon the 
second is a line || and equal to the line itself and ± to the trace of those planes. 

(2) A line obUque to one plane and || to a ± plane has its view upon the first a line shorter 
than the line itself and [ | to the trace of those planes; the Z the other view makes with the trace is 
equal to the Z the Une itself makes ■nath the first plane. 

(3) A line obhque to two _L planes has its views upon both shorter than the line itself and neither 
view shows the true length, nor true Zs the line itself makes with those planes. 

(4) The views of || hnes are || unless their views coincide, or are merel)- pts. 

(5) A surface || to a plane is ± to the X plane. Its view upon the first plane is a figure similar 
and equal to the surface itself. Its other view is a line || to the trace of the planes. 

(6) ^ surface obUque to one plane and ± to the ± plane is foreshortened 1 1 to the trace in its 
view upon the first. The Z the other view makes with the trace of those planes is equal to the Z 
the surface itself makes with the first plane. 

(7) The views of 1 1 and equal surfaces whose corresponding lines are 1 1 are similar figures whose 
corresponding lines are equal and ||, unless the \'iews coincide, or are merelj- lines. 

(8) If a line or figure be revolved from one position to another about an axis ± to a plane, its 
view upon that plane remains unchanged in form and dimensions. In the view upon a ± plane the 
distances of the pts. from the trace of the planes also remain unchanged. 



CHAPTER VI 
PLANE SECTIONS 

75. Principles and Methods. If a plane be imagined to intersect an object, 
the portion of the object lying within it is called a plane section, and the plane 
a cutting or section plane, denoted by C. P. 

(a) If the portion of the object hiding the section be imagined as removed, 
the view of the remaining portion or merely that of the section itself, upon a 
plane || to the C. P., is called a sectional view. A sectional view thus shows 
the sec. in its true shape. Such views are made in place of or in addition to 
external views to show the form, dimension, and arrangement of hidden parts 
more clearly; to aid in determining the views of irregular, oblique, and inter- 
secting forms; and to enable the draftsman to dispense with numerous dotted lines. 

The figures of this chapter show sees, of geometric forms; practical applica- 
tions are shown in Chap. XL 

(b) When a sec. is not 1| to V, H, or P, its true shape may be found by 
projecting upon an aux. plane || to the C. P., as in Art. 69; or by revolving the 
sec. II to V, H, or P, as in Art. 70. 

(c) The position of a C. P. is indicated in the view in which it is seen edge- 
wise; that is, by its trace upon V, H, or P. 

The section is generally indicated by drawing ||s across it called section 
lines, usually at 45° up to the right. The spacing is dependent upon the size 
and shape of the sec, usually tV", judged by eye. 

Avoid drawing the lines 1| to the main lines of a sec. 

When a drawing is to be inked, it is not desirable to pencil the sec. lines, 
but the sectioned portion may be indicated by a few lines sketched freehand. 
In views which show the true shape only, sec. lines may, for economy of time, 
be omitted. 

(d) In obtaining a sectional view the trace of the C. P. is first drawn. This 
determines the pts. in which the C. P. cuts the edges, elements, or other lines 
of the object. Next project these pts. to the corresponding lines in the other 
views, and join them. Then proceed as in (b). 

76. Objects Having Plane Surfaces. As the intersection of plane surfaces 
is a st. line, it follows that any plane sec. of an object bounded by plane surfaces, 
as a prism or pyramid, will be a polygon of 3, 4, or more sides according as the 
plane cuts 3, 4, or more surfaces. 

(a) A Prism. Fig. 138(a) represents a right square prism with its lateral 
faces at 45° with V, cut by a plane A-B, JL to its base and 1| to V. The pts. 
of intersection 1, 2, 3, and 4 with the top and bottom edges are, therefore, 
determined in the top view and then projected to the front. As the plane cuts 
four ± faces, the sec. is a rectangle; and, being || to V, is shown in its true shape 
in the front view. 

62 



PLANE SECTIONS 



63 



Fig. (b) represents the prism in the same position, cut by a plane _L to V, 
but oblique to H and P. The pts. of intersection are, therefore, determined 
in the front view. 

Fig. (c) represents the prism with its vert, faces at 30° and 60° with V, cut 
by a plane _L to P and oblique to V and H. This illustrates a case in which 
the sec. is not symmetrical with respect to a C. L. 




(b) A Pyramid. In Fig. 139 the C. P., D-E, cuts the lateral edges of the 
pyramid at 1, 2, 3, and 4. Pts. 1 and 3 may be projected to the top view in 
the usual manner; pts. 2 and 4, however, are in edges which are represented 
by vert, lines in both views. To obtain these pts. in the top view either 
of the follo^^^ng methods may be used. 

First Method. Draw a side view and transfer the distance 2^-4^ from it to 
the top view. 



64 



ESSENTIALS OF MECHANICAL DRAFTING 



Second Method. Imagine the edge A-B to be revolved i| to V to coincide 
with A-C. Then the front view of 2 will be at 2' and its top view at 2". If 
now the line A-B be rev6lved back to its original position, 2^ will be determined 
as shown; 4^ may in like manner be found. 

The C. P., F-G, gives a triangular sec. The method of finding pt. 7^ is pvident 
from the preceding. 

77. Objects Having Curved Surfaces. 

(a) A Sphere. Any sec. of a sphere is a O whose center is in the diam. 
of the sphere, _L to the C. P. See Art. 2(n). 

In Fig. 140 the top view of sec. on A-B is thus a O whose diam. is equal 
to 1^-2^. The front view of sec. on C-D is an ellipse. The intersection of the 




Fig. 139 




Fig. 140 



C P. with the hor. great O of the sphere determines 3 and 4 in the top view. 
As the front view of that O is in the hor. E-F, these pts. will be projected 
at 3^ and 4^, as shown. Intermediate pts. of the curve may be found by 
determining pts. of intersection of the C. P. with other Os of the sphere whose 
planes are || to either H or V. Thus, assuming a O parallel to V, cutting the 
O of the sec. at 5 and 6, these points will be determined in the top view and 
projected at 5^ and 6^ in the front. Again, assuming a hor. O to cut the sec. 
at 5 and 8, these pts. will be determined in the top view, and projected at 5^ 
and 8^. Note that the highest pt., 7, of the curve is that in which a hor. O 
becomes tangent to the plane of the sec. 

The assumed Os may be regarded as lines of intersection given by aux. C. Ps., 
each of which cuts the plarie of the sec. in a chord of both ©s, as 5-6, whose 
end pts. are thus in the required curve. 



PLANE SECTIONS 



65 



The intermediate pts. of sec. C-D may also be found by the reverse of the 
method of Art. 70. Thus the half sec. revolved || to H about diam. 3-4 
determines the distance of pts. 5 and 6 from diam. 3-4, to be set off above and 
below that line in the front view. 

(b) A Cylinder. Any sec. |1 to the axis of a right circular cyhnder is a 
rectangle, and a sec. || to the base, a O. See Art. 2(1). 

The sec. on A-B (Fig. 141) is an ellipse whose top view is a O coinciding with 
the view of the base. The sectional view may be obtained as in Art. 69(e), 
or by determining pts. of intersection of the C. P. with elements, as indicated. 

The assumed elements may be regarded as Knes of intersection given by aux. 
C. Ps. which cut the plane of the sec. in lines, as 2-6 and 3-5, whose ends are thus 
in the required curve. 

(c) A Cone. A sec. through the vertex and base of a right circular cone is 
a A; a sec. _L to the axis, a O whose center is in the axis of the cone. See 
Art. 2(m). 

The obUque sec. (Fig. 142) is an ellipse, pts.- for the top and sectional views 
of which may be obtained by determining the pts. of intersection with elements 
of the cone. A more accurate method is to assume a number of Os of the cone 
to cut the sec. These will be || hors. in the front view and Os concentric with 
the O of the base in the top. Thus, drawing the front view of a O through 
any pt., as 1^,- 3^, and projecting from 1^, 3^, to the top view of the O, pts. 
1^ and 3^ will be determined. 

The vert. sec. being _L to both V and H has both front and top views a st. 
line, but being || to P will show its true shape, a hj^perbola, on that plane. 

The C. P. cuts the right element at 2 and the O of the base at 4 and 5, which 
pts. may be obtained in the sectional view in the usual manner. Intermediate 
pts. maj^ be obtained by use of elements, but it is more accurate to assume other 
Os of the cone, to intersect the sec. Thus a O cutting the sec. at 6 and 7 
determines the distances of those pts. from the C. L. in the top view. 

The assumed elements and Os may be regarded as lines given by aux. C. Ps., 
as in the case of the cylinder, Art. (b). 




CHAPTER VII 



INTERSECTION OF SURFACES 



Fig 143 



78. Principles and Methods. The Hne or lines in which the surface of one 
integral part of an object intersects that of another part is called a line of iJiter- 
section. This line joins the pts. in which the edges, elements, or other lines of 
each part meet the surface of the other, as is evident from Figs. 144(a), 145(a). 
The problem, therefore, in determining the line of intersection is to find the 
pts. of intersection of the edges, or of a sufficient number of other lines, with 
the surfaces, and to draw the line through the pts. thus found. 

(a) The pt. of intersection of a line and a surface is determined in the view 
in which the surface is seen edgewise or as a line, and then projected to the other 
view of the line precisely as in finding the pts. of a plane section. 

Thus, in Fig. 143, line A-B intersects the front face of the object at 1, as 
determined in the side view, for if it be assumed to intersect the left side face, 

pt. C would be seen in the side view to come 
above the object. A-B also intersects the 
right side face at 2, as determined in the 
front view. 

When the intersected surface is not shown 

as a line in either of the required views, an 

airx. view which will so represent it may 

sometimes be used. The pt. of intersection 

of any line and surface may be determined 

by passing a plane through the line, ± to 

the plane of either of its views. This C. P. 

cuts the given surface in a line in which the 

required pt. of intersection must lie, for both 

lines are in the same plane. 

Thus, in Fig. 143, a plane ± to H and P gives a sec. 3 4 5 6, and as the lines 

of the sec. are in the surface of the pyramid, pts. 1 and 2, in which A-B intersects 

them, must be the required pts. Similarly a plane ± to V, gives sec. 7 8 9 10, by 

which 1 and 2 are determined in the top, or side view. 

(b) It is evident that a plane which cuts both intersecting parts of an object 
would give a plane sec. of each, and the pts., if any, in which the lines of these 
sees, intersect, being in the surfaces of both parts, must be pts. in the required 
Une of intersection. By assuming a series of such C. Ps., any desired number of 
pts. may be found. 

The position of the C. P. should, when possible, be so chosen that the sec. 
of each part is one readily obtained. 

(c) In problems involving intersections, first draw each part as complete as 
possible in itself, that is, without regard to its intersection. Next obtain the 

66 





INTERSECTION OF SURFACES 



67 



pts. of intersection of the edges or elements which can be directly determined 
in either of the views; then those whose pts. must be found by means of C.Ps., 
or aux. views. In finishing, only such lines should be rendered as edges or 
outlines as represent the edges or outlines of the object as a whole. 

79. Objects Having Plane Surfaces. Fig. 144 represents the intersection 
of a triangular prism (A) and an oblique hexagonal prism (B). The lateral 
edges, A-B, C-D, E-F, etc., of (B) intersect the left vert, surface of (A) at 1, 
3, 5, 7, 9, 11. As this surface is seen as a line upon both V and H, the pts. are 
determined in both views, and since all the pts. lie in the same plane as that 
of the surface, the line of intersection is simply the outline of a plane sec. of 
(B). In this case it is an irregular hexagon which would be seen in a side view. 

K" I" r^l2''IO' 

! Ti" 




Fig. 144 



The edge A-B intersects the upper surface of (A) also at 2. As the surface 
is seen as a surface upon H and as a line upon V, pt. 2 is determined in the front 
view and then projected to the top. Edges C-D, E-F, I-J, and K-L intersect 
the front and rear surfaces of (A) also, at 4, 6, 10, and 12, respectively. As 
these surfaces are seen as lines upon H, the pts. are there determined and then 
projected to the front view. Edge G-H and the right vert, edge of (A) are in 
the same plane || to V and thus seen to intersect in the front view at 8. 

Points 13 and 14 in which the hor. edges of (A) intersect the surfaces of (B) 
are not directly determined in either view, for neither of the intersected surfaces 
is there seen as a hne. Bj' obtaining an aux. view upon a plane _L to the axis 
of the oblique prism, the intersected surfaces of the latter will be seen as lines 
D -B and B*-L^ and the pts. of intersection of the edges of (A) determined 
at 13 and 14^ as shown. These pts. may then be projected to the front view. 
It is obviously unnecessary to draw the complete aux. view. 



68 



ESSENTIALS OF MECHANICAL DRAFTING 



When the preceding method is not convenient or possible, the pts. may be 
found by means of a C. P.; thus, if the plane of the hor. surface containing the 
edges be assumed to intersect (B) it would give a sec, the intersections of whose 
sides with the edges of the hor. surface determine 13 and 14 in the top view 
as shown. Again, if the plane of the front surface of (A) be assumed to intersect 
(B) it will give a sec, the intersection of a side of which with the front hor. edge 




Fig. 145 



of (A) determines pt. 13 in the front view. Pt. 14 could in like manner be found. 
It is evident that merely those lines or portions of the sees, which determine 
the required pts. need be drawn. 



80. Objects Having Curved Surfaces. 




Fig. 146 



(a) Fig. 145 represents the inter- 
section of two circular cylinders (A) 
and (B) and an equilateral triangular 
prism (C). The line of intersection 
of (A) and (B) is determined by the 
pts. of intersection of the elements 
of one cylinder with the surface of 
the other. Thus B, D, F, and H 
in which the upper, front, lower, and 
rear elements of (B) intersect (A) are 
determined in the top view, for the 
cylindric surface of (A) is there seen 
as a hne (circle). 

Intermediate pts. may be found 
by assuming intermediate elements 
of (A) in a side view, or of (B) as 
shown; the positions of the elements 



INTERSECTION OF SURFACES 



69 



of the latter being transferred from top to front view bj^ means of the revolved 
half-base, Art. 73. 

Note also that C. Ps. || to the axes of the cylinders would cut elements of 
both, which intersect in pts. of the required curve. 

The views of prism (C) may be determined by means of a side view, or a 
revolved half-base as shown. The pts. of intersection of the hor. edges with 
cyhnder (A) are determined in the top view. Note that the rear surface inter- 
sects the cylinder in an element. The inclined surfaces intersect (A) in elliptic 
curves, intermediate points in which may be found by assuming elements of 
the cyhnder and determining their intersection with (C) in a side view; or by 
assuming lines upon the prism || to its axis. Thus a hor., 5-6, intersects the 
cylinder at 5, determined in the top view. 





Again, a C. P. || to the axes of (A) and (C) would give an element of (A) 
*nd two lines of (C) whose pts. of intersection must be in the required curves 
as shown. Note that pts. 9 and 11, in which the direction of curvature changes, 
are at the intersection of lines 9-10 and 11-12 with the right element of (A). 

(b) In determining the curve of intersection of cjdinders (A) and (B) in Fig. 
146, the same methods would be used as in (A) and (B) of Fig. 145. 

Equidistant elements of (B) may be determined by an obUque aux. view, or a 
revolved half-sec. taken J. to the axis, as shown. 

(c) In determining the intersections in Fig. 147, the C. Ps. may be taken 
horizontallj', or through elements, as shown. If cylinder (c) were oblique, an 
oblique aux. view could be used. 

(d) The simplest method of determining the intersection in Fig. 148 is to 
pass C. Ps. through the elements of (B). The sees, of (B) will thus be As and 
those of (A) ellipses, or other curves. 

(e) Fig. 149 represents a form similar to the stub end of a connecting rod. 
The highest pts., B and D, of the curves, in which the bell-shaped neck of the 



70 



ESSENTIALS OF MECHANICAL DRAFTING 



cylindric portion intersects the rectangular, are directly determined in the front 
and side views, and will be seen from the plan to be the pts. in which circular 
sees, become tangent to the sides of rectangular sees. The lowest pts., A, C, 
E, and F, lie in the O through the corners of the rectangle and, therefore, located 
in the front and side views by projecting from the plan to the corresponding 
views of that O , which are obtained as shown. 

Other pts. are determined by intermediate G. Ps. Thus plane G-H gives 
a O which cuts the rectangle at 1, 2, 3, and 4 of the required curve, which are 
located in the other views as in the preceding. 




Fig. 149 



CHAPTER VIII 

DEVELOPMENT OF SURFACES 

81. Principles and Methods. A development is the representation of the 
surfaces of an object as laid out, unfolded, or unrolled into the plane of the draw- 
ing. The operation is suggested pictorially by Figs. 150 (a), (b). A develop- 
ment thus shows the exact area of all surfaces of the object and the exact length 
of every hne of those surfaces, Fig. 151. Plane, cylindric, and conic surfaces 
only can thus be developed. Surfaces of double curvature and warped surfaces 
may be developed approximately by assuming portions to be cylindric, conic, 
or triangular,. 




Fig, 150 




Developments are made to determine the shapes of surface patterns required 
in constructing objects of sheet metal, cardboard, etc.; to plot groove outlines 
for cylindric cams; and to obtain templets or patterns for irregular surfaces, etc. 

(a) To OBTAIN A DEVELOPMENT. First draw the views from which the 
measurements of the lines can be made. The surface may be imagined as opened 
on any lines but its different parts should, so far as practicable, be represented 

• as attached to each other, — each being so placed that if the dev. were cut out 
upon its outer line and properly bent, it would form or envelope the object, 
surface for surface, line for line. 

In obtaining lengths from the views, transfer the measurements with dividers. 

Remember that a line shows its true length only upon a plane to which it 
•is ||. If not thus shown, its true length must first be determined. The dev. may 
be placed in any convenient position. In some cases it is possible to place it 
so that one set of dimensions may be projected, as in Fig. 151. It is sometimes 
desirable to attach it to some line or lines of a view (Figs. 154, 159-161), prac- 
tically as revolved about those lines. 

(b) The reproduction of forms in thick paper, by cutting and bending devs., 
will prove an excellent aid to correct solutions. The surfaces may be held in 
position by providing paste laps (see Fig. 151) or by using gummed binding. 
To obtain neat edges the folding lines should be lightly scored with a sharp- 
pointed knife. 

71 



72 



ESSENTIALS OF MECHANICAL DRAFTING 



(c) In practical work, allowance must be made for seams, laps, thickness 
of material, etc. Economy in cutting is also important. 

Material for double curved, or warped surfaces, is cut to patterns of assumed 
cylindric, conic, or triangular portions and then beaten, pressed, or stretched to 
the required form. In some cases the material is stamped by dies, or spun to 
form in a lathe. 

82. Objects Having Plane Surfaces. 

(a) Peisms. The dev. of a right prism consists of two similar polygons for 
the bases and as many rectangles as the prism has sides. In obtaining the dev. 
of the hexagonal prism (Fig. 151), observe that the lateral edges are || to V, and 
the base edges to H. The front and rear base edges are also 1 1 to V. 

First set off the true lengths of the lower base edges upon a line, A-A. At 
pts. A, B, C, etc., draw _Ls. Upon these set off the lengths of the vert, edges 
and connect the pts. to complete the lateral surface. Attach the bases to any side. 



7"r" : 6"E" 




Fig 151 



To develop the prism after the portion above an oblique C. P. 1-4, has 
been removed, obtain base line A-A, lower base, and _Ls as in preceding. Now 
assuming the surface to be opened from edge A-1, this edge will be the outer 
verts. A-1, in the development. Beginning with A-1, transfer the lengths of 
the verts, to the corresponding lines in the dev. Similarly transfer lengths 
of base edges G-4 and G-5 to the hor. through G. The lines joining 1, 2, 3, 4, 
and 5, 6, 7, 1 will be the dev. of the line of intersection of the C. P. with the 
lateral surface, and must, therefore, agree in length with the corresponding lines 
of the sectional view. Such lines should always be compared with dividers. 
The method of obtaining the portion G4'5 of the upper base is evident. To 
complete the dev., draw a figure similar and equal to the true shape of the oblique 
surface by Art. 56, attaching it to any side, as B C 3 2. 



DE^ELOP^IE^"T of surfaces 73 

(b) Pyramids. The dev. of a right pj-ramid consists of a polygon for the 
base and as many As as the pyramid has sides. The slant edges of the rect- 
angular pyramid (Fig. 152) are equal, but are not projected in their true length 
in either view. Assuming the surface to be opened from one of these, obtain its 
true length as in Art. 71, and with this as rad., describe an indefinite arc DEBCD. 
Upon the arc set off the successive lengths of the base edges. ' Join these pts. 
and each to the center A, the vertex, to complete the lateral surface. Then 
attach the base. Or, lay off first one side of the base as E-B, and with E and 
B as centers and the true length of the slant edges as rad., describe arcs to inter- 
sect at A; then proceed as in preceding. 

Since the altitude of each side is ± to a base edge, the vertex A could be 
determined by setting off the true length of an altitude, as A-F, ± to its corre- 
sponding base edge E-B, as indicated. 

In pyramids whose inclined edges are not equal, it is necessary to obtain 
the true length of each separately, and to construct the As by Art. 37, joining 
them on their common sides. 




BT'f 



To develop the pyramid (Fig. 152) when truncated, first obtain dev. of entire 
pyramid. Then obtain the true distances of pts. 1, 2, 3, and 4, from the vei'tex, 
or base (Art. 71). Set off these distances upon the corresponding fines in the 
dev. and join the pts. thus found to complete the sides. Finally, copy the true 
shape of the top surface from the sectional view, attaching it to any side, as 
BC21. 

83. Objects Having CyUndric or Conic Surfaces. 

(a) Cylinders. Since the elements of a right circular cylinder are equal 
and _L to the bases, the dev. of the curved surface (Fig. 153) will be a rectangle 
whose height is equal to the length of the elements, and base equal to the length 
of the circumference. 

The length of the base edge may be obtained as in Art. 32, and laid off 



74 



ESSENTIALS OF MECHANICAL DRAFTING 



Fig 153 




upon a line A-A. At the ends of this hne draw J_s and complete the rectangle. 
As each of the bases would touch the dev. of the curved surface at but one 
pt., it is not necessary to attach them. 

To develop the curved surface when cut off on line 12 3 4 1, obtain base 
line A-A, divide it into 12 or 24 equal parts according to the number of elements 
assumed upon the cylinder, and through the pts. draw _Ls. The length of the 
cut elements may then be transferred from the front view as in the case of the 
edges of a prism. The curve traced through 1, 2, 3, 4, 1 will complete the lateral 
surface. See Art. 18(a). 

When, as in (B) Fig. 159, neither end is a O, both will develop as curves. 
It is necessarjr, therefore, to assume some O of the surface whose development 



Pattern for A and A', 
)r one-half of E.. 




Fig. 1 54 



Pattern for F U.5e right half for B, 
/left half for B' 
»J/ DETAIL OF 

PILASTER CAP 

Make 4 16 Oz Copper 



DEVELOPMENT OF SURFACES 




may be used as a base line 
upon which to set off the true 
distances between the ele- 
ments and from which to set 
off their lengths. Such line 
may be obtained by a C. P., 
a-E^, taken _L to the ele- 
ments as shown. See 
Fig. 160. 

Fig. 154 illustrates 
dev. of an object with 
lindric curved sides, 
true distances between 
elements of surfaces 
and (B) are determined 
upon A-E and L-0 in the 
elevation and their lengths in 

the plan. The distances between the elements of (E) and (F) are identical with 
those of (A) and (B). To determine the true distances between the elements 
of (C) and (D) it is necessary, since these elements are not ± to V, H, or P, 
to obtain the profiles of (C) and (D) by aux. views or sees, upon planes ± to the 
elements, as indicated. 

(b) Cones. Since the elements of a right circular cone are of equal length, 
the dev. of the curved surface (Fig. 155) will be a sector whose radial sides, A-D, 
are equal to the true length of an element and whose arc, D-D, equals the length 
of the circumference of the base. To determine the length of this arc, obtain 
iV or TT of the base circumference and set it off 12 or 24 times in the dev. 

To develop the portion below the parabolic sec, obtain the base arc D-D, 
divide it into 12 or 24 equal parts according to the number of elements assumed 
upon the cone and draw the elements. Then find their lengths when cut off, 
as in the case of the pj-ramid. Art. 82(b). 

Since the left element is 1| to V, the true distance of pt. 1 from the vertex 
A is seen in the front view. The true distances from A of intermediate pts., 
as 2 and 5, are not seen in either view, but as the elements of the cone are equal, 
hors. from these pts. to A-B will determine upon the latter the required distances. 
(See Art. 71(b).) Having transferred these to corresponding elements in the 
dev., trace the curve through them. 

The method of developing a right circular conic surface whose ends are not 
in planes J_ to its axis is evident from Fig. 161(b). 

In a conic surface other than right circular as in Fig. 156, or one whose vertex 
is inaccessible as in Fig. 161, it is necessary to assume the surface to be composed 
of plane As whose sides are elements and whose bases are short chords of the base 
of the cone. The method, called development by triangulation, is thus identical 
with that used in developing a pyramid with unequal slant edges. Art. 82(b). 

To develop the oblique elliptic cone (Fig. 156) obtain equidistant elements 
as shown. Assuming the surface to be opened upon A-D, the line A-B will be the 



76 



ESSENTIALS OF MECHANICAL DRAFTING 



C. L. of the dev. and may be drawn directly at A'-B', equal to its true length 
A^-B^. The true lengths of elements A-1, A-2, etc., may be found by revolving 
them 1 1 to V. To avoid confusing the views, however, a separate diagram of true 
lengths may be constructed. The true length of any element will be the hypot- 
enuse of a right A whose base is equal to the length of that element in the plan 
and its altitude equal to the vert, height of one end pt. above the other in the ele- 




FiG 156 



Fig. 157 




vation. Hence, draw A-X J. to the base line and equal to the altitude of the cone; 
then lay off X-1, X-2, etc., equal to A^-1^, A^-2''^, etc., and connect pts. 1, 2, 
etc., to A. Observe that this is equivalent to revolving the elements 1 1 to P. Now 
with A' as center and rad. A-1 draw an arc across A'-B'; with B' as center and 
rad. B^-1^ intersect this arc at 1'. Then A'-l' will be the developed position 
of A-1. Continue this operation until all elements have been determined and 
trace the curve B'1'2', etc., through the base pts. as indicated. 

The figure also illustrates the method of developing the cone when truncated. 



DEVELOPMENT OF SURFACES 



77 



84. Objects Having Double Curved, or Warped Surfaces. 

(a) Fig. 157 illustrates the method of developing a double curved surface 
by assuming portions, as A 7 8, to be cylindric, also by assuming portions between 
the O s to be conic. 

Fig. 158 illustrates a transition piece of piping, A, whose surface is warped, 
being neither cylindric nor conic. Such surfaces can be developed approxi- 
mately only, by triangulation. Art. 83(b). 




Fig 156 

Half Pa+te-m for A rm f i i- • i i • . 

ihe surface may be divided into 
measurable As as shown. In the plan 
divide half of the base and curved end 
of the top edge into the same number 
of equal parts, at least six. Observe 
that as the planes of both edges are ob- 
lique to H, each must first be revolved 
II to H, or V, and then back to its 
oblique position. Join the corre- 
sponding pts. in each with full lines 
1-A, 2-B, etc.; also draw lines 2-A, 3-B, 
etc., dotted for contl-ast. 

Next construct the diagram (a) for 
the true lengths of the full lines, noting 

that neither the upper nor lower pts. are on the same levels. Similarly construct 

diagram (b) of true lengths for the dotted lines. 

As the surface is symmetrical about the C. L. 1-Y, the hne 1-X may be drawn 

directly at I'-X' in the dev. (c), equal to its true length 1^-X^. With 1' as 




78 



ESSENTIALS OF MECHANICAL DRAFTING 



center and rad. 1-A of diagram (a) describe an arc across I'-X'. With X' as 
center and rad. X-A of the plan, intersect this arc at A'. Then I'-A' will be the 
developed position of line 1-A. With A' as center and rad. A-2 of diagram (b), 
describe an arc. With 1' as center and rad. 1-2 of the revolved base, intersect 
this arc at 2'. Then A'-2' will be the developed position of line A-2. 

With 2' as center and rad. 2-B of diagram (a), describe an arc. With A' 
as center and rad. A-B of the revolved top edge, intersect this arc at B'. Then 
2'-B' will be the developed position of 2-B. Continue these operations until 
all points have been determined and complete the dev. as shown. 

85. Intersecting Surfaces. In intersecting parts of an object, the line of 
intersection being common to both surfaces will be developed in each. As 
this line is determined by the pts. in which the edges, elements, or other lines 




Fig. 159 



of each part intersect the surfaces of the other, and the dev. gives the true length 
of every line of every surface, it follows that these pts. may be found by trans- 
ferring the true lengths of the lines which determine them in the views, to the 
corresponding lines drawn in the dev. 

(a) The lateral surface of (B) (Fig. 159) is developed as explained in Art. 
83(a). The dev. of the rear half only is shown and is attached upon element 
A-B for convenience in transferring lengths. The distances between the elements 
are seen in the revolved sec. of a-E^ and transferred to the base line E-a, as shown. 

(b) The method of developing the lateral surface of (C) is evident. Lines 
as 1-2 and 3-4, which give intermediate pts. of the line of intersection, are found 
by obtaining their true distances from the || edges in the revolved sec, and 
their true lengths, from either the front or top view. The dev. of the upper 
half only is shown. 



DE\TELOPMENT OF SURFACES 



79 



(c) To develop the lateral surface of (A), first obtain the dev. of the entire 
surface, that is, a rectangle (rear half only is shown). Assume surface to be 
opened from right element. This element cuts the surface of (C) in I and M. 
The opposite element cuts the surface of (B) in F and B and will be in the center 
of the dev. From these elements the positions of the other elements of (A) 




which cut the surfaces of (B) and (C) may be determined. The distances of 
these elements from the outer elements are seen in the top view, and the distances 
of their points of intersection from the bases of (A) are seen in the front view. 
(d) Fig. 160 illustrates a pipe elbow with conic and pyramidal flanges; Fig. 
161, two other common forms, the methods of developing which are indicated. 



riG. 161 




CHAPTER IX 

MECHANICAL PICTORIAL DRAWINGS 

86. Character and Purpose of the Drawing. It has been noted that two views 
at least are required to show the exact form, size, and relation of all lines and 
surfaces of an object, and that two only of its three dimensions can be shown 
in their actual proportion and relation in one view. It is sometimes necessary, 
however, to represent an object or detail by a single oblique view having a more 
or less pictorial effect, while at the same time showing the relative proportions 
of its principal lines or dimensions to a scale. Compare Figs. 162 (a), (b), (c), 
etc.; 163 (a) and (b); 169 (a) and (b). 




(a) 



F D 

ORTHOGRAPHIC 
PROJECTION 



Fig. 162 



(b) 



(c) 



(d) 



G 




^ 








& 


B 


J^ 
^ 


UN ^ 




^ A 


i 


\ 


' ^ 


F 






1 


~~~^ 


^-^ 




1 


D F 


C 




F D 



ISOMETRIC 
PROJECTION 



OBLIQUE PROJECTIONS 



f- 


-^ 


c 


c 


A 




1 




D 

TRUE PICTORIAL REPRESENTATIONS 



Such representations are made in place of or supplementary to the usual 
views for the purpose of general illustration, or to describe details of machine, 
furniture, and building construction and assembly more directly and clearly. 
They are also extensively employed for catalog illustrations and Patent Office 
drawings. 

The methods ordinarily used are those of Isometric Projection and Oblique 
Projection, the choice depending upon the form of the object and the particular 
effect desired to be given. A drawing made by either method is never a true 
picture, since || lines are always represented by Ijs as in orthographic proj., 



80 



MECHANICAL PICTORIAL DRAWINGS 81 

while in a true pictorial representation the apparent convergence of receding 
l|s is shown. Hence, such drawings often give an unsatisfactory effect of 
distortion. 

87. Isometric Projection. If a cube, placed as shown in Fig. 162(a), be 
turned 45° about a vert, axis, then forward until the diagonal of the cube 
through A is ± to V, all of the edges will be equally inclined to that plane, and 
being equal in length their projs. upon it will be equal. Such view is an isometric 
(equal measure) projection of the cube. (See Fig. (b).) 

When a rectangular object is thus related to the plane of proj., it will be 
seen that each of its edges will be || to one or another of three mutually _L lines, 
as A-B, A-C, and A-D, which correspond in direction to the axes or principal 
dimensions (1, b, and t) of the object. These lines are called the isometric axes, 
and their projs. form equal Zs of 120° about a pt. A. All lines of the object 
coinciding with or || to the isometric axes are called isometric lines; all others are 
non-isometric lines. The planes determined by the isometric axes and all planes 
II to these are called isometric planes. 

Note that isometric lines onl}', make equal Zs with the plane of proj.; hence, 
equal distances upon such lines only will be equally foreshortened in the view. 

(a) Instead of obtaining the view by the methods of Art. 72, an isometric 
is usually constructed by setting ofT the limiting pts. of the lines directly upon 
isom. lines and joining the points thus determined. Measurements must never 
be made upon non-isometric lines. The foreshortening of the isometric lines, 
which is about .81 of full size, is usually disregarded and the measurements 
made equal to the true lengths of the corresponding lines of the object, or to 
some other common scale. 

One of the axes is usually assumed as vert.; the others, therefore, are at 30° 
with the hor. direction, as in Fig. 162(b). When the lower surface of an object 
is to be visible the axes are reversed. Invisible lines are omitted unless they 
give necessary information. 

When dimensions are given, the dimension and extension lines must be 
isometric. See Fig. 163. 

In general, sections and breaks should be taken in isom. planes. See Figs. 
164, 170, 194; also invisible section. Fig. 165. 

88. To draw the isometric of a rectangular object. 

(a) A Cube. Fig. 162(b). From any pt., as A, draw indefinite lines A-B 
and A-C at 30° and a vert. A-D. Upon these set off the given dimensions of 
the object to the desired scale. Since each edge is || to one or another of the 
isom. axes, it will be || to the corresponding line in the drawing, and the rep- 
resentation may be completed as shown. It is evident that the isom. could be 
started from any assumed pt., as D or E. 

(b) An Object with Rectangular Details. Fig. 163. Starting at any 
pt., as A, draw the isom. of the bottom board (A), obtaining all measurements 
from the front and top views. 



82 



ESSENTIALS OF MECHANICAL DRAFTING 



n" 



F e 



JA 4- 



1-14 



G -lot 






CM tnIS 

LI 



■0*2 1 




Fig 163 






F 




i3 
B 


y 


I 

1 


1 
1 


\ 


i 

c 


2 


_\ 






Fig. 165 



MECHANICAL PICTORIAL DRAWINGS 



83 




Details must be built up from the surface or surfaces which they intersect. 
Thus to draw block (B), locate a corner, as E, and draw the line of intersection 
E 1 2 3; then proceed practically as with (A). 

To draw (C), one of its lower corners, as F, must first be located upon the 
top face of (A) by means of co-ordinates B-4 and 4-F; that is, by measuring the 
distance of F from B along 

lines which will be || to \ / (n) 

two of the isom. axes. To 
draw the recess in the front 
of (C), locate a corner, 
as G, by co-ordinates F-5 
and 5-G, and- proceed as 
before. Corner H is de- 
termined in like manner. 

89. To draw the iso- 
metric of an object involv- 
ing non-isometric figures. 
The form and position of an integral part of an object are frequently such that 
some or all of its lines will be non-isom. Such lines are determined in the isom. 
by co-ordinates || to two or to all three of the axes. 

(a) Straight Lines. In Fig. 165 each of the inclined lines in (a) will be 
oblique to two of the axes and hence determined by co-ordinates 1| to those 
axes, as shown. See also oblique hues in Figs. 171, 172, 194. 

In Fig. 166 the hne 
A-B M'ill be oblique to 
the three axes. The end 
B, was located in this case 
by co-ordinates A-1, 1-2, 
2-B II to those axes. The 
end pts. of C-D were de- 
termined in Uke manner. 

(b) Polygons. In Fig. 
167 two sides only of the 
hexagon ABCDEF wiU be 
isom. Each pt. may be 
referred by co-ordinates 
to two of the isom. axes, 
as in (a), or to the sides 
to those axes. 

By placing this rectangle with one of its lines |1 to or coincident with its 
isom., one set of dimensions may be projected to the required figure, as shown. 

(c) Curves. The method of determining a curve in an isom. drawing is 
identical with that of a rectilinear figure save that, in the absence of vertices, 
pts. must be assumed in the curve and referred to the axes, as illustrated in 
Figs. 168(a), 169, 170. 




of a circumscribing rectangle whose sides will be 



84 



ESSENTIALS OF MECHANICAL DRAFTING 



The projection of a O in an isom. plane is an ellipse, in determining which 
it is convenient to circumscribe a square. Fig. 168(a). As the axes of the ellipse 
coincide with the diagonals of the isom. square, their end pts. 8, 6, 5, and 7 could 
be determined by co-ordinates and the curve described by Art. 64(b). See 
also Art. 64(d), Note 2. 





Fig 168 " 

An approximate method of drawing the ellipse by circular arcs, which is 
usually sufficiently exact, is shown in Fig. 168(b). The centers are the inter- 
sections of _Ls to the sides of the square at the middle pts. 

The application of this method to the rounding of corners is shown in Fig. 
169. The construction at D determines the radii for all. 

To draw a curve not in an isom plane, as A-B (Fig. 170), determining pts. 
must be located by co-ordinates || to the three axes, as indicated. Screw 
threads are usually represented conventionally as in Fig. 173. 




Fig. 169 



(d) Solids. The method of drawing an object by inscribing it in an isom. 
solid is evident from Figs. 167, 171. When it cannot be thus inscribed, pts. 
must be referred to the axes as indicated in Fig. 166. 

In order to preserve the symmetrical appearance of the piece in Fig. 172, 
it was necessary, either to draw the top view turned through 45°, or to take the 
hor. measurements for the groove centers on 45° lines instead of on hors. and verts., 



MECHAXICAL PICTORIAL DRAWINGS 



85 





'A 



■+M-- 



FiG 171 





Fig. 172 



86 



ESSENTIALS OF MECHANICAL DRAFTING 



as shown. Observe that the outer elements of conic surfaces would be tangent 
to the ellipses and not drawn to ends of the axes. See A and B, Fig. 173. 

To determine the outlines of surfaces of double curvature, obtain a series 
of sections and draw the required curve tangent to the isom. of these, as indicated 
in Figs. 174, 175, also 149. 

90. Oblique Projection. An oblique projection is obtained by means of 
II projectors oblique to the plane, the object being so placed that two of its 
axes or principal dimensions are || to the plane. In the case of a rectangular 
object, as a cube, the view, therefore, gives the true shape and size of its front 
and rear faces and two dimensions of the object. All lines ± to the plane are 
projected as || lines whose direction and lengths depend upon the direction of 
the projectors. They are usually drawn at 30°, 45°, or 60°, up or down to left 




Fig 173 




or right, and made equal to the full scale length of the corresponding lines of 
the object as in Fig. 176, or foreshortened, usually half, as in Fig. 177. The 
latter gives a better pictorial effect, but requires the use of two scales, as is evident. 

When the receding ±s are foreshortened one-half and inclined at 45°, the 
view is sometimes called a cabinet projection. 

The axes in oblique proj. are thus a vert. A-B, a hor. A-C, and a line A-D 
at 30°, 45°, or other Z . Measurements must be made only upon these axes 
or lines || to them. Curves, and lines not || to the axes, must be determined 
by co-ordinates || to the axes, as in isom. drawing. 

A circle not || to the plane of proj. will be projected as an ellipse, which may 
be obtained as in Art. 89(c). 



MECHANICAL PICTORIAL DRAWINGS 



87 



An obvious advantage of oblique proj. over isom. lies in the possibility of 
representing certain curved and irregular surfaces in their exact shape and size. 
See applications of principles in Figs. 178, 181. 




Fig 175 




Fig. 176 



Fig. 177 




Fig. 178 



91. Shade Lines, Shadow Lines, and Line Shading. In finishing, visible 
edges between light and dark surfaces are frequently shaded, as in Figs. 162(b), 
164, 167, 169, 178(a). The indication of these shade lines adds relief to the 
drawing and increases its pictorial effect. 



88 ESSENTIALS OF MECHANICAL DRAFTING 

In isometric drawing the rays of light are assumed 1 1 to the plane of proj . and 
at 30° down to the right. Hence, all rectangular objects and parts, whose lines 
are || to the axes, have their shade lines in the same relative positions as in the 
cube. Fig. 162(b). 

In oblique proj. the rays are assumed || to the diagonal, C-E, of the cube. 
Fig. 162(c). 

In determining the shade lines, the shadows are disregarded. Elements of 
curved surfaces are generally not shaded. Edges of cylindric surfaces may be 
shaded as in Figs. 169, 178(a). 

Instead of shade lines, shadow lines may be applied as in Art. 23. (See 
Figs. 171, 178(b).) Another method is to shade the nearest edges as in Fig. 172. 
Line shading may be applied as in Art. 24. 



CHAPTER X 
WORKING DRAWINGS 

92. Character and Purpose of the Drawing. A working drawing is a mechan- 
ical drawing, containing all information as to form, dimensions, construction, 
material, finish, etc., necessary to the workman or mechanic in making or 
building the object which it represents. See Art. 1. 

To convey this information readily, the drawing must be accurate; as 
clear, simple, and direct as possible; and in accordance with shop and drafting 
practice. It must express the idea completely and definitely and contain noth- 
ing that is unnecessary, ambiguous, or misleading. In commercial drafting 
utility of the drawing and economy of production are the ends sought for in 
all cases. 

The nature of the views of which the drawing is composed is explained in 
Chap. V. 

93. Types of Drawings. In complicated objects composed of different 
pieces or parts, certain features would inevitably be hidden or not clearly shown; 
hence, in such cases, two types of drawings are required — namely, general or 
assembly drawings and detail drawings. 

(a) The purpose of a general drawing (Figs. 179, 184, 185, 202) is to illustrate 
the design of the subject as a whole and to show the relative positions of the 
different pieces composing it. It may include the complete description of some 
or of all of the pieces, or give merely such information as may be necessary in 
assembling them or erecting the object. As a rule, it should be as free from 
representation of minor detail and hidden parts as possible. 

(b) A detail drawing (Figs. 180, 203-206,210-213) shows each piece by itself 
and gives all information necessary for making it. In simple objects, full 
instructions would ordinarily be given in a general drawing, sometimes called 
a detailed assembly. See Figs. 179, 184, 185. 

Note. — The table shown in Fig 179 was separately detailed (Fig. ISO) for purpose of comparison 
with the general drawing and to illustrate certain methods of arrangement, dimensioning, etc. 

Detail drawings may be shown on the same sheet with the assembly, grouped 
on separate sheets, or each piece shown on a sheet by itself, depending upon 
the character of the object, size and number of parts, etc. In general, details 
of the pieces of one part of an object should be grouped apart from similar groups 
of other parts (Fig. 205) ; and so far as possible the arrangement of details of 
related pieces should be such that reference may readily be made from one to 
the other. Figs. 180, 205. It is often desirable to show related pieces of a 
part as assembled. Figs. 181, 182. 

In drawings of machinery the special information required by the different 
workmen, as the pattern maker, the blacksmith, and the machinist, is often 

89 



90 



ESSENTIALS OF MECHANICAL DRAFTING 



! 





o 
a. 

r 

H 

z 
o 

I- 
o 



U 



£0 


t 


!<■ 


II 

-IN 


>■ 




DC 


fll 


< 


7^ 


rr 


o 


CQ 


</; 



-^^l- 



?ikls^ 



r!~~r 



a 



nl 



Lz!z:r 



:^j 



-?De— 

-191- 



-— 4»t'-|-Tl'* 



^^^^^^^^^^^^^g^ 



■^ 



Fig. 179 



WORIvING DRAWINGS 



91 



-I- "jH- Q O 




i s 

_J 



0) - 



< o 



to ''^ 
-J _] 
< • 



4- 4 



III! 



k-Ti-^ 



— |s — -f 

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I \ 1 









^ in 5 



1 
I 
J_.J 







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?SI- 



-^62- 



tJ_i. 



M 



a: 

IS ; 
Id -i 
_1 - 

£« 

-I 
li- ■^ 



Fig. 180 



92 



ESSENTIALS OF MECHANICAL DRAFTING 



detailed upon separate sheets for each. Likewise work required to be done on 
certain machines, etc., is often so grouped. 

94. Position of Object and Arrangement of Views. 

(a) The object should be represented in its natural position, and, when it 
has a definite front, that part should be shown in the front or principal view. 
The position of a separate piece, however, should be such as to show best its 
form, with the fewest lines and views, regardless of its position in the complete 
object. 

(b) In general, a top, side, bottom, or rear view should be related to the 
front view, as indicated in Fig. 120. When for convenience or necessity they 
are not thus arranged, it is well to letter them: "Top View," "Side View," etc. 
See Figs. 181, 192. Other views may be placed where most convenient, the 
part to which each refers being clearly indicated by its position or by marking. 
See Figs. 179, 182. 



TT+F 




DETAILS OF 

DRAWER 



Scale: 
Make 



\n 



i^ Front Q.S W.Oak 

"raw. Stock Plain Oak 



3-Ply, About 4^ 




95. Selection and Number of Views, etc. Those views should be selected 
for drawing which most clearly and adequately describe the object and require 
the least time to execute. Views that do not add clearness or convey necessary 
information should be omitted. 

Two or three views are ordinarily required; others, however, are frequently 
necessary. In simple symmetrical objects one is often ^ufRcient. Fig. 174(a). 

In addition to the external views the following are frequently necessary: — 

Sectional views — to show interior construction. 

Diagrams — to show the direction of motion of moving parts, relations of 
important centers, etc. 

Developments — to show true shapes for surface patterns, templets, etc. 

Isometric, or oblique views — to show details of construction, etc., with a 
pictorial effect. 



WORiaNG DRAWINGS 



93 



Fig. 182 



PILLOW BLOCK 
AT 45° 




! l._J 1 [A 

I I i i ' 



.|U-^-|-"Drin 13 Holes at^"= 9' 
. 1 0" 



i h 



Fig. 183 




— B 

1} 

i 



o 



^"Drill, 12 Holes 



|"Drill, 12 Holes 
Equally Spaced 



94 



ESSENTIALS OF MECHANICAL DRAFTING 



96. Center Lines. As a rule, symmetrical objects and parts should have 
their axes and centers indicated by means of center lines. See Art. 66(g). 
As the object is usually represented with its main axes |1 to the planes of the 
views, the main or principal C. Ls. of a drawing are usually vert, and hor. lines 
passing through the centers of the corresponding views of the main body of 
the object. These C. Ls. are commonly extended to connect the views. 

Secondary C. Ls. are likewise usually _L to each other or to main C. Ls., 
but are ordinarily not extended. When the centers of a series of symmetrical 
parts are equidistant from a common center, a circular C. L. is used. The other 
C. Ls. for these parts are usually radial lines from the center of the circular C. L. 
Figs. 182-184, 188, 189. 

When a C. L. coincides with a line of the object, the latter should be shown. 
See Figs. 184, 220. 

A straight C. L. may be regarded as the edge view of a center plane. 




FLANGE COUPLING 



Fig. 184 



97. Conventional Representations. Instead of true or complete projections 
it is often desirable, for clearness or economy of labor, to make conventional or 
approximate representations. Some of the more common methods used are 
referred to in the following: — 

Partial views, as of one-half or other suggestive portion of an object, may 
often be used in place of complete views. Figs. 180, 182(a), 184, 188(a), 192-195, 
203-206. 

Lines or details clearly shown in one or two views may often be omitted 
in others, especially those of hidden parts. Figs. 184, 192, 193, 199. 

Of a series of similar parts, as holes, bolts, etc., of the same size, it is usually 
necessary to draw but one or two and to indicate the locations of the others. 
A brief note often saves the drawing of many lines. Fig. 183. 

Screw and pipe threads, bolts and springs, are usually represented conven- 
tionally, as in Chapter XI. 




^ik-fs— J 




Pig. 185 



96 



ESSENTIALS OF MECHANICAL DRAFTING 



Ellipses and other non-circular curves may often be approximated by arcs 
ofOs. Art. 18(b). 

Where an edge is rounded so that no definite line of intersection is seen, it 
is often better to show such line as if existing. See curve a-b, Figs. 188, 208. 

In Fig. 184, instead of projecting the upper bolt from the circular view, it 
is represented at its true distance from the center of the shaft, the same as the 
lower bolt; thus avoiding confusion of the view and 
suggesting the symmetry of the piece. For similar' 
reasons the lower arm of the wheel (Fig. 188) would 
be shown as in (b), the same as the vert, arm., instead 
of foreshortened as in (c). See also grooves in Fig. 
172(a), and slots and ribs in Fig. 189. 

Other conventions are described in Arts. 98 and 99. pi^ 155 

98. Sectional Views^ See Chap. VI. 

(a) When a section does not lie in a main center plane, the place where the 
sec. is taken should be indicated, as in Figs. 179, 188(a), 192, 199. Parts lying 
beyond the sec. need not be shown unless they add clearness or give addi- 
tional information. When lines of a removed portion are shown they may 
be indicated as construction lines. See top view, Figs. 179, 193, 199. 

Section lines should have the same direction and spacing throughout all 
parts of a sec. of any one piece. Fig. 188(b). Different pieces in a sec. are 
indicated by sec. lining in opposite directions, at different Zs, or by difference 
in spacing. Figs. 184, 185, 192. Sec. lines must never cross figures, arrowheads, 
or notes placed in a sec. See Fig. 184. 





CAST IRON 



WROT IRON STEEL CASTING WRO'T STEEL 




^ 






BABBITT. ETC 



RUBBER. ETC 



BRICK WOOD 

Fig 187 



</<//> 


AA/' 


</v// 


•y////' 


•^///y/ 


'///''/, 


// // /i 



BRASS, ETC. 




STONE. 




CONCRETE 



When a sec. is very narrow, it is sometimes filled-in black, and adjacent pieces 
separated by narrow spaces. Fig. 186(a). Very short sec. lines may be drawn 
freehand. In lai-ge drawings, sees, are often indicated as in Fig. 186(b). 

(b) Material Conventions. Different kinds of materials may be indicated 
by different kinds of sec. lining. The conventions shown in Fig. 187 are com- 
monly employed, but there is no fixed standard of practice. Many draftsmen 
use plain sec. lining for all materials and indicate the kind by lettering. On 
drawings finished on paper the sees, are sometimes tinted with India ink or colors. 



WORKING DRAWINGS 



97 



(c) Selection of Sectional Views. A view may show a complete section, 
that is, through the entire object; or a partial section. In objects symmetrical 
about an axis, half only on either side of the C. L. need be sectioned. Figs. 
189(b), 192. When a partial sec. is not limited bj' a C. L. or some line of the 
object it is usually shown as in Figs. 199, 218(o). 

A view need not show all parts in sec. that lie within the sec. plane, unless 
the drawing is rendered clearer, or additional information given. In general a 
solid cylindric part as a shaft, rod, bolt, screw, etc., intersected by a plane || 
to its axis, should be shown in full^ likewise a key, rib, tooth, wheel arm, or 
turned handle. 




Fig. 188 



In Fig. 188(a) the plane A-A passes through the rim, vert, arm, hub, and 
handle of a wheel. Instead of making a true sec. (c), as projected from the 
view (a), the draftsman would make the conventional sec. (b), in which the vert. 
arm and handle are shown in full. He would also represent the lower arm as 
in the same center plane as the vert. arm. If considered by itself, the true sec. 
would suggest a solid web between the hub and rim, which would be misleading, 
while the conventional sec. shows the desired information at a glance. 

The conventional sec. of a ribbed piece is shown in Fig. 189(b). The method 
(c) is also used. Note location of slots in (b) and (c). 

It is frequently desirable to show two or more || sees, in the same sectional 
view. See Fig. 190. 

A sec. may sometimes be shown as revolved upon the part sectioned. See 
sec. on D-D, Fig. 188(a), also 185, 198. In such case the original view is drawn 



98 



ESSENTIALS OF MECHANICAL DRAFTING 



complete and the sec. in full or dashed lines. It is generally better to "break" 
the view and show the sec. in the space. 

Instead of making a separate sec. view, an invisible section may be indicated, 
as in Figs. 182, 208. 





Fig. 169 

99. Broken Views. When only a portion of an object is required to be 
shown and the portion is not limited by C. Ls., the views may be broken, as 
indicated in Figs. 191, 179, 180, 181, 203, 204. The outline of the break is some- 
times omitted. Fig. 188(a). 

The broken ends of symmetrical pieces may often be made to suggest the 
sectional shape. 







I" 



I I 
I I 






■10" 




Tig. 190 



ROUND BAR 





z 







FLAT BAR WOOD 

Fig. 191 



WORKING DRAWINGS 



99 



100. Standard Sizes of Sheets and Scale of Drawings. 

(a) For convenience in handling, filing, etc., the drawings are usually made 
upon sheets cut to standard sizes, which are peculiar to each shop or office and 
dependent largelj^ upon the uses for which the drawings are intended. For 
sizes in common use, see Art. 4. 



HALF PlAn 
Showing Details at Top. 



HALF SECTION Ion A-B 




h ^5^^* k-3i-^ 



PEDESTAL 

Scale: 3"= Iff. 



Fig. 192 

(b) All drawings are made to a definite scale. Art. 11. The scale chosen 
must be such that all parts and dimensions will be shown clearly. Details 
should be to full size, if practicable; small details to an enlarged scale, if nec- 
essary. It is desirable, however, that all drawings on a sheet be to the same 
scale. 



100 



ESSENTIALS OF MECHANICAL DRAFTING 



Scales commonly used are full size, and |, \, |, y^i t^' A' ^^'^ tf ^i^^- 
These are usually stated on the drawing thus: Scale: Full Size. Scale: Half 
Size. Scale: 3" = 1 ft. Scale: li" = l ft., etc. 

(c) To determine the scale necessary to be used for a given size of sheet, 
find the ratio of the available space to the full size dimensions, making due allow- 
ance for spaces between the views and from margins. Thus, supposing the 
drawing space horizontally to be 13|" minus 2" for spaces, and the hor. di- 
mensions 20" plus 31", the ratio would be 11| to 23| and the nearest convenient 



Top removed 




TABORET 

Scale; 3" = I ft. 



Fig. 193 



scale, considering the hor. dimensions and spaces only, would, therefore, be half 
size or 6" = 1 ft. To aid in such calculations, the layout sketch would be 
used. Art. 105(a). 

101. Dimensioning. Although the drawing is made to a definite stated 
scale, generally, the accuracy of the views themselves is not depended upon 
to indicate the size of the object even when drawn full size; all dimensions 
required by the workman must be shown upon the drawing in figures, or other- 
wise definitely specified. 

In order to give the essential dimensions and to omit those that would be 
unnecessary, misleading, or impractical, the draftsman must consider the needs 
and convenience of the workman, and the successive steps and processes involved 



WORKING DRAWINGS 



101 




Fig. 194 



in the construction of each part. In addition to a knowledge of shop require- 
ments, he must exercise good judgment in deciding where to place the dimensions, 
so that they can be easily found and apphed. This article describes the methods 
of dimensioning for commonly occurring cases. 

(a) Forms of Dimensions. To allow for the use of a two-foot rule in 
working, ordinary dimensions less than two feet are usually given in inches 
and halves, 4th, 8ths, etc.; thus, 17/;^". Others are given in feet and inches; thus, 
2'-0", 2 '-6", 2'-0i"; or thus, 2 ft. 0", 2 ft. 6", 2 ft. Qa". Some offices use inches 




Fig. 195 



102 



ESSENTIALS OF MECHANICAL DRAFTING 



up to 36, others up to 72, and in some classes of work all dimensions are given 
in inches. When the greatest possible exactness is required, dimensions must 
be given in decimal form; thus, 0".94:, 2". 442. When all dimensions are under- 
stood to be in inches, the sign (") may be omitted. Limits of allowable variation 
in size are often indicated thus, Tllfl:, which means that the measurement 
must not be greater than 1.500 nor less than 1.498. 



(a) 



'fOicO 



r 



"- J. 



I' 



-32 



5 
*I6 



13" 
'32* 



3" 
"4' 



|n^32 r^r*F^x' 
— ■ " ,g 



32 



u-)T<5 



I" 



3 



'8 



5" 
132 



toiS 



(C) 



27 



32 



2 

Finish 

15" 



(b) 



7" 



8 



^oi^ 



16 



19" 



32 

1 



rs^¥:^ 



'v5° 





-l^f- 



Taper 



f per ft. 



I — 

'4 



Fig. 196 



(b) Dimension Lines, Figures, and Arrowheads. The figures are placed 
upon dimension lines. These lines, with the exception of those for radial dimen- 
sions and unlimited distances, have arrowheads at both ends. For forms and 
sizes of figures and arrowheads, see Figs. 73, 74. Arrowheads must touch the 
lines between which the dimension is given, and the figure must state the full size 
of the corresponding measurement of the object regardless of the scale of the 
drawing. Acceptable methods of placing figures and arrowheads are shown in 
Figs. 196, 197. 



WORKING DRAWINGS 



103 



Figures should be so placed that they can easilj^ be read from lower and 
right sides of the drawing. Avoid placing a figure where it will interfere with 
others, or with other lines. 

It is not permissible to place the figures for more than one dimension between 
the same arrowheads, nor upon other than dimension lines. 




Fig. 197 



(c) Locations of Dimensions. Dimension lines for linear measurements 
must always be ± to the parallels between which the dimensions are given, and 
ordinarily not nearer than |" to object lines and other dimension hues, C. Ls., 
etc. They should not cross each other or any line of the views, nor be drawn 
as continuations of other lines, if avoidable. In general, place dimensions 
outside of views, unless greater clearness and ease in reading will result by placing 



104 ESSENTIALS OF MECHANICAL DRAFTING 

them otherwise. Dimensions may be extended beyond parts dimensioned by 
means of extension lines. Pointers may also be used. Figs. 196(a), 201. 

Never dimension a distance in a view in which it is foreshortened. Dimen- 
sion a detail preferably in a view in which it is visible. So far as practicable 
give related dimensions, as of the length; width, and location of a hole, in the 
same view. Dimensions clearly given in one view should not be repeated in 
another upon the same sheet. When parts are obviously alike, dimension one 
or two only. 

Distances should be given from lines which represent finished or trued 
surfaces, and from C. Ls. Do not give distances from C. Ls. when not necessary. 
Locate symmeti'ical parts, in general, by giving distances to their C. Ls. Figs. 
196, 197, 183, 184. In a series of such parts, give distances between centers. 
As a rule, give dimensions of successive distances in the same direction from 
the same surface or C. L. The final dimension of a series is sometimes omitted 
to indicate this surface or line more definitely. Give total or overall dimensions 
as well as all detail or sub-dimensions, so that the workman will not be obliged 
to calculate. 

Place II dimension lines in the order of their length, the longest farthest 
from the part dimensioned, to avoid crossing the dimension lines and causing 
confusion. Figs. 180, 196, 198. 

(d) Angles and Tapers. Lines appearing to be hor., vert., or _L to each 
other are assumed to be so unless otherwise dimensioned. An Z may be 
dimensioned by co-ordinate dimension lines, or by an arc described from its 
vertex as center. Fig. 196(c). See also 171(a). Common methods of dimen- 
sioning tapers are shown in Fig. 196(d). Short tapers are sometimes dimensioned 
in degrees. Standard tapers are specified by number and kind; as, "No. 2 Morse 
Taper," etc. 

(e) Circles and Arcs. In general, give the diam. dimension of a O and 
place the figure on an oblique diam., or extend the dimension to the most con- 
venient location. Fig. 197(a). Small Os may be dimensioned as in Fig. (b). 
Give the rad. of an arc with an arrowhead at the curve end only. When 
the space is too small for the figure, dimension as in (c). Radii |" or less, 
of fillets and finish arcs, ordinarily need not be given. When the rad. is known 
or unimportant, indicate by Rad. or R. When holes or bolts are arranged in a 
O , give diam. or rad. of the circular C. L. When equal spacing is not evident, 
specify by note "Equally Spaced." If unequal, dimension as indicated in Fig. 
197(b). 

(f) Irregular or Non-Circular Curves. These may usually be dimen- 
sioned by giving the lengths and positions of offsets J. to appropriate base lines. 
Fig. 198, also 174(a). In most cases, however, it is more practical to omit such 
dimensions*, and to provide the workman with one or more exact patterns or 
templets of thin material against which he can lay out the curves directly on 
the piece. 



*Fig. 198 is thus dimensioned merely to illustrate the method. 



WORKING DBAWINGS 



105 



(g) Round, Square, Hexagonal, and Octagonal Pieces. When the cir- 
cular, square, etc., shape is not shown in any view, nor specified by note or title, 
indicate by Dia. or D., Sq., Hex., or Oct., after the diameter dimension. Figs. 
174(a), 199. 

(h) Standard Measurements. 

Screws, pipes, bolts, and springs: give dimensions as in Chap. XI. 

Tubing: give outside diam. and thickness by gage, or in thousandths of 
an inch. 

Wire: give diam. by gage, or in thousandths of an inch. 

Sheet Metal: give thickness by gage, or in thousandths of an inch. 

(i) Assembly Drawings. The dimensions required depend largely upon 
the kind of object and the purpose of the drawing. In machine assemblies 
usuallj^ only the important overall dimensions and locations of principal C. I.s. 
are necessary. 



Fig 198 




102. Lettering. 

(a) Explanatory Notes, etc. Specifications and directions concerning the 
kind of material to be used, the name and number wanted of each piece, kind 
of finish to be given, kind of fit required, and any other necessary information 
not shown by the drawing or stated in a title or bill of material, must be expressed 
in brief, concise notes lettered upon the sheet. See Art. 25. Notes should 
preferably be located outside of the views and to read horizontally, or vertically 
from the bottom. The part noted may be indicated by a pointer. Fig. 188. 

Finished Surfaces. When a surface of a casting or forging is to be machined 
or finished by filing, turning, milling, grinding, etc., an allowance must be made 
on the piece for this finish. Ordinary finish is generally indicated by a letter f, 
placed on the surface in all views which show the surface as a line. Fig. 196, also 



106 



ESSENTIALS OF MECHANICAL DRAFTING 



189 (a), (b). The f s should be placed to read horizontally as shown. When the 
piece is to be finished all over the i's are omitted, and the note "Finish all over" 
or "F. A. O." placed near the principal view. The limits of a finished portion 



Top Removed 




SECTION AT A-B. 



\^zk- 







Q..S.W.Oak. 



r-J-1 



1 



-12 



■I7sq.- 



10- 



m 



* - 1 -- 



T 



J_ 



-16- 




JARDINIERE STAND 

Scale; 3" = I ft. 



Fig. 199 



may be indicated as in Fig. 196(a), or by a note. It is often necessary to specify 
the kind of finish; as "Polished," "Ground," "Milled," "Reamed," etc. In this 
case the f's are also usually omitted. 

When a portion of an unfinished surface is to be machined to form a bearing 



WORKING DRAWINGS 



107 




Fig. 200 



for a bolt-head, nut, stud-shoulder, etc., it may be specified 
thus, "Spot Face,"or"Counterbore to Surface," according as an 
allowance is, or is not, to be provided for this finish. 

Knurled surfaces are indicated by the word "Knurl," or as 
in Fig. 200. The varying spaces and Z s of the lines are esti- 
mated by ej'e. 

Circular Holes. "Cored," "Drilled," "Countersunk," etc., 
holes should be specified in one view, or the other, as in Fig. 201; "Tapped" holes 
as in Fig. 218. 

Ki?id of Fit. When one metal piece is to be fitted into another, the fit is 
specified as "Running, "Drive," "Force," 
or "Shrink" Fit; or by indicating the 
limits of variation in the sizes of the parts. 
See Art. 101(a). 

Treatment of Metal. Special treatment 
is specified, as "Tempered," "Hardened," 
"Case Hardened," "Blued," etc. 

Standard Parts. Parts such as screws, 
bolts, keys, etc., which conform in design 
and dimension to recognized commercial 
standards are usually omitted from the 
detail drawings and specified by note, or 
Usted in a bill of material. Art. (d). 

(b) Identifying Marks. It is cus- 
tomary to give each piece of a machine 
a distinguishing number or letter under 
which it is Usted and referred to. A cor- 
responding mark is placed on the drawing 
of the piece and is often accompanied by 
its name. Figs. 202-206, 210-213. 

(c) Titles. For convenience in filing, 
etc., the title of the drawing is generally 
placed in the lower right corner. Figs. 
179-180, 202-206. The title should des- 
ignate: — 

1. The name of the object, part, or 
particular detail shown, or all three. 

2. The scale, if uniform. 

3. The date of completion of the 
drawing. 

4. The draftsman's signature. 
If a single detail only is shown, the 

number required and material are gener- 
ally stated. Some or all of the following 
information is also generally included: 
The type of drawing, — as assembly, detail, Fi g. 20 1 




Drill- 



108 ESSENTIALS OF MECHANICAL DRAFTING 

shop, etc.; the signatures of the tracer and persons by whom the drawing is 
checked and approved; the firm for whom the drawing is made; references to 
other drawings, etc. 

Sub-titles. On a sheet of details each piece may have a title, consisting of 
its name and identifying mark; the scale, if not stated in the main title; number 
required if more than one; material; pattern number, if a casting; and finish, 
if to be all over. If a bill of material is given, the identifying mark and the 
scale only are necessarJ^ 

Planning a Title. First write the title upon a separate paper. Having 
decided upon the statements to be included, and the size and style of lettering, 
proceed to draw the vert. C. L. of the title space and the guide lines for the 
heights. Next, upon another paper and adjacent to its upper edge, compose 
and sketch the first line of lettering to the size to be used upon the drawing. 
Then, placing this sketch against the proper guide line, with the middle point 
of the line of letters at the C. L., point off the widths of the letters and draw 
each carefully. Proceed in like manner with other lines of lettering. 

(d) Filing Index and Bill of Material. A filing index giving the num- 
ber or other designating mark of the machine and the sheet number is usually 
placed in the lower right corner or included in the main title. It is often shown 
in an upper corner also. Figs. 202-206, 210-213. 

In a set of drawings, the assembly may be indexed as sheet 1, and may include 
a list of the other drawings with their numbers. The sheet number of each 
detail may be given near its identifying mark; thus, (3-5): the 3 indicating the 
mark, and the 5 the sheet no. containing the detail. ' 

A sheet of details is usually accompanied by a hill of material placed above 
or at the left of the main title, accounting for each piece and giving its mark, 
name, material, no. wanted, and other necessary description and information. 
Por material to be cut to size, the rough stock dimensions should be specified. 

When the number of parts is large or more than one sheet of details is neces- 
sary, a separate bill grouping the forgings, castings, etc., and giving the sheet 
xio. of each part, may be made. Fig. 213. 

(e) Abbreviations. The following are some of those in common use: — 
R. H. Right-hand S. Steel 0. H. S. Open Hearth Steel 
L. H. Left-hand M. S. Machine Steel . C. H. S. Case Hardened Steel 
C. I. Cast Iron T. S. Tool Steel S. C. Steel Casting 

W.I. Wrought Iron C. R. S. Cold Rolled Steel Bz. Bronze 

103. Shadow Lining and Line Shading. Arts. 23, 24. Shadow lines and line 
shading are used when the advantage gained in clearness and effect is sufficient 
to warrant the expenditure of the time necessary to apply them, otherwise they 
are omitted. They are of special value on drawings of complicated objects 
shown by few views or whose corresponding views are upon different sheets, 
as in some assembly drawings. They are rarely used on ordinary detail or 
shop drawings. 

104. Sketching. To the designer, draftsman, or mechanic, skill in making 
freehand sketches for the rapid expression of ideas of exact form and structure, 



1-oi-ns 




riG 202 



50| 




—di) 



> 



3 




=12 
in 



////////////////////////////////, 



lO'" SPEED LATHE 
ASSEMBLY 

DAY MF(3 CO BOSTON 
Scolo 6-.|Ft Dat« 1-29-IS 

Dr ^i»^ Tr ^g^^ Ch.T^f-^ ^p.A^ 



SL-IO- 



WORKING DRAWINGS 109 

and as an aid in the solution of constructive problems, is of great importance. 
To the student, careful sketching is fully as valuable, as a means of acquiring 
ability to make and to read working drawings, as instrumental drawing. 

(a) In making working sketches from objects the following order should be 
observed : — 

1. Separate the parts, if necessary, and sketch the views of each in detail. 
Begin with the main C. Ls. and principal surfaces, blocking in first the larger 
details of the piece and proceeding in like manner down to the minor details. 
See Fig. 209 (a), (b), (c). 

Distances and directions are generally determined by eye. Co-ordinate 
paper, ruled in I" or I" squares, is often used and affords a more ready and 
accurate means of obtaining the desired proportions, etc. For use of pencil 
see Art. 9(b). 

The number and character of the views should be such as to express all 
required facts clearly. Leave nothing to memory. Make the sketches large 
enough to prevent crowding the notes and figures. The sketches should be 
intelligible to any one familiar with working drawings, and should enable the 
scale drawings to be readily made from them without having to resort to the 
object. Isometric or oblique views may often be used to advantage. 

2. Indicate the finished surfaces and put on the necessary extension lines, 
dimension lines, and arrowheads. 

3. Obtain the dimensions by careful measurement of the object and put 
on the explanatory notes. 

Each piece should be dimensioned independently of the others. 

(b) Measuring Objects. In taking measurements, a foot, or a two-foot 
rule may be used for ordinary work and steel rules, gages, etc., for fine work. 
Obtain distances from trued or finished surfaces whenever possible. In obtain- 
ing inside and outside diams., calipers may be used. 

In an object of varjang diams. (Fig. 174(a)), measure the diams. at a sufficient 
number of pts. and locate these diams. by measurements || to the axes. In 
measuring an irregular form, as the table leg (Fig. 198), it is necessary to es- 
tablish a base line, as A-B, by means of a triangle or a carpenter's square, 
and to determine the lengths of ± offsets to it with a foot rule 

In locating a circular hole, measure to edge of hole and add half its diam. 

In ordinary finished surfaces, take the nearest 32d; in rough work, the 
nearest 16th; in finely finished objects and parts, absolute exactness is necessary. 

105. Making Scale Drawings. 

(a) Having completed the sketches, decide upon the number and arrange- 
ment of the views to be shown in the drawing, and the necessary scale. To 
aid in determining the scale and the locations of the chosen views upon the 
sheet, a rough layout sketch indicating the general outlines, main C. Ls., and 
margins, should be made. Thus, from the layout (Fig. 207) for the bearing 
shown in Fig. 208, it will be seen that the minimum space required horizontally, 
not including spaces between views and from margins, will be 2a + 2b. Simi- 
larly, that required verticalh' will be c + d + 2b. From these two sets of 




Fig. 203 




Fig. 204 




rie.205 





1 




1 , ; ' 




\r 


' 




1 

L 




-> =1 




f% 


H* 


/; 


d 






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^*r: 


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< 

It: 




t 






< 








t 


s 






E iH^ 


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)-*« 


-HO 


^ 




T 


^, 


'1 






Oin 






o 




ftjVy 


I 


- 


-1 








■ 


o 








:\\ 


0- 




nw- 


J__ 




V 


V 


^-I 1 


I:.ij 


\ 


I-* 








\ 


1 




r 

o 

I 

en 




Fio. 206 



114 



ESSENTIALS OF MECHANICAL DRAFTING 









, 










1 


» 

-x-4- \ 


. J . 















\. a J 






*b-i4^b^ 








7 " 




) [ 






I 1 




T 


3 


^ J 






; 








I 




, 1 








1 






1 




1 1 

LAYOUT SKETCH 
\ FOR PLAIN BEARING 


TITLE SPACE 












Fig 


207 













5^ Core. 5 Deep 




Cast Iron Patt*B-5 



FiG.208 



PLAIN BEARING 

Scale: Full Size Date- 3-7-18 
Dr ^e. Tr K/E.. Ch. AP. App. CF. 

M-I8-I 



WORKING DRAWINGS 



115 



dimensions, with due allowance for spaces and title, the scale maybe determined, 
as in Art. 100(c). Having decided this, the dimensions A, B, C, and D for 
the locations of the C. Ls., should be estimated, proper allowance being made 
where necessary to preserve a well-balanced sheet. 

The same general directions should be observed in all drawings, as lack of 
provisions for the necessary views, etc., may lead to errors impossible to rectify 
without re-drawing the entire sheet. 

(b) Having completed the layout, the scale drawing (Fig. 208) may be begun. 
Four stages in penciling are indicated in Fig. 209. For general working instruc- 
tions, see Art. .3. In drawing from objects, the detail drawings are first made 
and the general drawing built up, piece by piece, from them. In designing 




four stages in penciling 
Fig 209 

or planning objects to be built, the general drawing is usually first begun and 
the detail drawings worked out from it. In some cases it is necessary to carry 
along both detail and general drawings at the same time. 

106. Tracing and Blue-printing. 

(a) Instead of inking the pencil drawmg, tne finished drawing is usually 
obtained by tracing in ink upon tracing Hnen or paper fastened over the original. 
This tracing is then filed as the permanent record of the construction and used 
for making blue-prints or other copies for shop and general use. 

The original is sometimes penciled directly upon the tracing material. 



C/) 

r 
j_ 

o 
I 




Fig. 210 






[ 





-fl — 






-Hf 


- 








^. 






KT*. 









y-J 












V* 


ai? 
























X 






c 






o /^ 


^) 


i 

o 

X 

u 
d 


— 




N 


X 


-1 \, 

g 

ll. 

J 
-1 

g 

=12 


J 

LO 
to 














< 






'■ 1 






XX 




-?* 








^ 


c- 


^1 












H> 



Vi 



:^ 



Tie 211 




riG.212 




I 
r 


1 -1 




- -4— l-N N 

''J 








I 



I 




j/TUMQim/) 



616 ( 



5^r 



t- 

D 
Z 

uj D 



O "1 

<< < 



'(0,(0, to (O 



X.X I _1 c ? 



< 5s < 



a. U,V- 
< *i-l < lU 



5 






4i 



. r.j n <- 1.-1 i^ i^ cr- lT, o - PJ 
3,0,0 O O O O O C -1 



Fig. 213 



120 ESSENTIALS OF MECHANICAL DRAFTING 

The dull side of tracing linen takes the ink better, but erasures can be made 
more readily upon the glazed side. Either will take the ink more readily if 
rubbed with a cloth and chalk powder. Penciling should be done on the dull 
side. Carry but little ink in the pen, and make the lines somewhat wider than 
ordinary as the lines print finer than those of the tracing. 

As the linen contracts and expands unevenly, tracings that cannot be com- 
pleted the same day should be inked by sections. 

Make ink erasures with hard eraser, rubbing gently to avoid injury to the 
surface. Restore the smoothness by rubbing with soapstone or a smooth piece 
of bone. Pencil lines may be erased with soft eraser. 

(b) Blue-prints are usually taken in a printing frame having a glass front 
and removable backboard. The tracing, or original drawing if on translucent 
material, is placed with the drawing side next to the glass and its under side 
in contact with the chemical coated surface of blue-print paper. The glazed 
side of the frame is then exposed to direct sunlight, which, penetrating the part 
of the tracing material not covered by the ink lines, causes the chemical not 
thus protected to change color and to adhere permanently to the paper, while 
that under the lines remains unchanged. After suitable exposure the paper is 
removed from the frame and soaked in water, which dissolves the unfixed 
chemical, leaving white lines on a blue ground. 

107. Checking Drawings. It is customary not to permit a drawing to be 
worked from until it has been checked by a careful, systematic examination, 
and approved by the head draftsman. In checking, it is well to assume every- 
thing to be incorrect until proved to the contrary. The following order may 
be observed: — 

See that each piece has been represented and that its views are properly related. 

Check views of each piece for correct and adequate description of form and 
-construction. 

Note if C. Ls. and all necessary dimensions and notes are given. 

Scale every dimension, and verify overall dimensions by computation: 

Compare the figures on all parts that are to fit together. 

Check measurements in details and assembly and note if they agree. 

Finally, see that all items required to be recorded in the title and bill of 
material are complete and correct. 

108. Reading Drawings. Ability to read working drawings rapidly and 
intelligently is quite as important as skill in making them. This ability can 
be acquired through the study of such drawings and comparison with the objects 
represented; through the execution of drawings from objects; through the 
making of mechanical pictorial drawings and developments, from good examples; 
and by making the objects, that is, working from drawings. 

In reading a drawing, first fix in mind the general shape of the main body 
of the object, observing if the outline shows it to be rectangular, cylindric, etc., 
or a modification of such forms. Then observe modifications of the general 
shape, proceeding from the more important details down to the minor details. 

Note carefully the conventional methods employed in the representation 
and complete mentally the graphic statement of what is required. 



WORKING DRAWINGS 121 

Endeavor to visualize the object; to see in each view, not mere lines, 
but the object itself as if standing out of the paper. Regard the front view 
as the object directly in front of you; in looking at the top view imagine your- 
self looking down upon the object; in looking at the side view imagine yourself 
as viewing the object in a direction at right Zs to the front, and so on. 

Finally, note the dimensions, and specifications as to materials, finish, etc. 
All information as to sizes should be obtained from the figured dimensions and 
specifications; rarely by measuring the drawing itself. 

It is evident that full knowledge of the form, size, and relation of the lines 
and surfaces of each part of the object represented can be secured only through 
the information shown in all the views taken together. 



CHAPTER XI 



HELICAL CURVES, THREADED PARTS, AND SPRINGS 



lOQ. Helices. If a pt., A (Fig. 214) be imagined to move along the generating 
line, A-12, of a surface of revolution, while the line itself revolves about the axis, 
the pt. will generate a line of double curvature called a helix. 

The distance that the pt. advances, measured |j to the axis, during one 
revolution of the line, is called the pitch of the helix. 

If the rate of motion of the pt. and generating line of a cylindric helix be 
uniform, that is, if the pt. advances J, ^, or other fractional part of the pitch 
distance during the same fractional part of a turn, the helix is uniform or 
equable; if otherwise, it is variable. 

Again the curve is a right-hand or left-hand helix, according as the generating 
pt. rises to the right or to the left in the front half of a turn when the axis is vert. 

The helix has many applications in mechanical and architectural design, 
notably in screw threads, some forms of springs, winding stair rails, etc. 

(a) To DRAW A RIGHT-HAND EQUABLE CYLINDRIC HELIX. Fig. 214. AsSUme 

and represent equidistant positions of the generating element, as 1, 2, 3, etc. 
Divide the pitch distance, as A-A^^, into the same number of equal parts and 
draw hors. through the pts. of division to cut the elements at A^, A^, A^, etc. 
Assuming A to be the generating pt., A^, A^, A^, etc., will be twelve pts. of the 
desired curve, for evidently when the genei'ating ele- 
ment has made tV of a revolution the advancing pt. 
A will have moved tV of the pitch distance and will, 
therefore, be in the element 1 at A^. When the element 
has moved | a revolution the pt. A will have moved 
I of the pitch distance and will be in the element 6 
at A®, and so on. 

As the curvature is more abrupt at the outer 
elements, additional pts. should be determined by 
sub-division as shown. Having fixed the pts. with 
the needle, trace the helix through them freehand. 



Fig. 214 



DEVELOPMENT OF HELIX 



All 





CYLINDRl|C HELIX 



122 



HELICAL CURVES, THREADED PARTS, AND SPRINGS 123 

Points for other turns of the curve may be located by stepping off the pitch 
distance upon the elements, from the pts. of the first turn. The manner of 
obtaining parallel or double helices is evident; also the method of drawing 
conic or other helices. 

(b) In ruling and inking the helix observe directions given in Art. 18(a). 
"When several cylindric helices of the same pitch are to be drawn, a templet 
of a half turn, as A-A®, may be made. 

(c) The development of a cylindric helix of one turn is the hypotenuse of 
a right A whose base is equal to the circumference and whose altitude equals 
the pitch, as shown. 

(d) Any desired motion of a pt. may be plotted in a development of a cylinder 
and then projected back to the view, as for example in designing a cam for 
converting circular into reciprocating motion. 

110. Screw Threads. If a cylindric bar be revolved at a uniform velocity 
upon its axis, in a lathe, and the point of a V-shaped cutting tool be pressed 
against its surface and moved at a uniform rate parallel to the axis, the tool 
will cut a helical groove, V-shaped in section. If this groove be cut so that 
a similar projecting portion is left upon the bar, a V screw thread will be formed. 
Fig. 215(a). 

Similarly, if a square groove be cut and a square projecting portion left upon 
the bar, a square screw thread will be formed. Fig. 215(c). 

Fig. (b) illustrates a sec. of a V-threaded and Fig. (d) of a square-threaded 
hole. Observe that the helical lines of the threads correspond to the invisible 
lines of the screws. 

The V thread is commonly used on screws for fastening purposes; the square 
thread generally on screws for transmitting motion in the direction of the axis. 
All other threads are modifications of the V and square forms. 

A screw thread is right- or left-hand according as its helices are right- or left- 
hand. Thus a right-hand screw would turn around to the right (clockwise) 
in advancing or entering the part into which it is inserted. 

The diameter of the top of the thread is called the nominal diameter of the 
screw. The diam. of the bottom of the thread is the root diameter. The distance 
from the root to the top of the thread measured J_ to the axis is the depth of 
the thread. The distance between the centers of adjacent threads measured 
II to the axis is called the pitch of the thread. The term "pitch" is often used 
to designate the no. of threads per inch; thus "14 pitch" means 14 threads to 
the inch. The distance that the screw would advance in one turn is called the 
lead of the screw. In a single thread screw the lead equals the pitch; in a double 
or triple thread it is two or three times the pitch. 

(a) Multiple Threads. Screws are generally right-hand and single thread 
as shown in Fig. 215 (a), (c). If the pitch of a screw having the diam. and thread 
sec. shown in Fig. (c) be made say two or three times as great, the increase in 
the depth would obviously be such as to weaken the screw at the root. There- 



124 



ESSENTIALS OF MECHANICAL DRAFTING 



V SCREW THREAD 

(o) y^r^V^'"'^ 



(c) 



A 



SQUARE SCREW THREAD 



jingle 



■^,t^^\\ 



(e) 



Double 




J-d-l— Roo-tj Dia^ I 
(b) U— Nomirial Dia.— -I 



d4— Root; Diar , 
- — Norn i rial DiO: — -| (d} 




L= Lead of Screw. 
P= Pitch of Thread 
d = Depth of Th'd 



Fig 215 




P = .24 VD + 62S- 175 
d=.65P F = | 

(d) ^— P-^, 



jSHARP V TH'D 
d= 866 P 






Tw^^jm^ 



7W//m^/^ 



y/my. 



SQUARE TH'D 
d= sP + oi W=5P 




P= OS D +.04 nearly 
d= 64 P 



D = Nominal Diam of Screw 
P = Pitch ^y, 

n = No of Threads per inch 
d = Depth of Th'd 



d= 5P + 01 
W= 3707 P- OOS2 



Fig 216 



HELICAL CURl'ES, THREADED PARTS, AND SPRINGS 



125 



fore, in designing a screw of which the lead shall be two or three times the pitch, 
instead of cutting a single thread, a second or third independent parallel thread 
would be cut. Such would be a double, or triple thread screw. 

Fig. (e) represents a right-hand double square thread of the same diam. and 
pitch as the single thread of Fig. (c). 

Observe that in a single thread screw, the top on one side is diametrically 
opposite the bottom on the other, while in a double thread the tops are opposite. 

(b) To DRAW A SCREW THREAD. The diam., pitch, and thread sec. being 
given, as in Fig. 215(a); first obtain the elevation and an end view, or half of 
revolved base of a cylinder whose diam. equals the nominal diam. of the screw. 
Lay off the pitch A-C and draw the sec. ADC. Then beginning at A, draw 
the helix A-B-C-, etc., for the top of the thread by Art. 110(a). To obtain 
the root helix, draw a semicircle in the base view concentric with the first, 
obtaining the rad. by projecting from D. Then beginning at D and using the 
same pitch, proceed as with the outer helix. Observe that the outlines of the 
thread come outside of the V sec. and are tangent to the helices. When the 
pitch is small, they practically coincide with the sec. outline. 

(c) Standard Proportions. In the preceding figures, the threads were 
represented with large pitch in order to show the construction more clearly. 
The proportions of the threads most commonly used and the formulse for obtain- 
ing them are given in Fig. 216. In this country the Z of the V thread is usually 
60°, but for general work the tops and bottoms are flattened as shown in Fig. (a). 

The following table gives the U. S. St'd no. of threads per inch, for diams. 
from i" to 4|". 



U. S. Standard Screw Threads 



Diam. 


Thds. 


Diam. 


Thds. 


Diam. 


Thds. 


Diam. 


Thds. 


Diam. 


Thds. 


Screw 


Per In. 


Screw 


Per In. 


Screw 


Per In. 


Screw 


Per In. 


Screw 


Per In. 


1 

4 


20 


5 


11 


1 


8 


If 


5 


3 


^ 


A 


18 


1 1 
1 6 


11 


1| 


7 


11 


5 


3i 


^ 


f 


16 


3 
1 


10 


n 


7 


2 


4* 


^ 


31 


TW 


14 


H 


10 


If 


6 


2i 


4| 


3i 


3 


1 

2 


13 


i 


9 


14 


6 


2i 


4 


4 


3 


n 

T7 


12 


1 


9 


If 


5| 


2| 


4 


4i 


2| 



On square threads the no. of threads per inch is commonly \ of the U. S. 
St'd. In drawing, the depth is made f • 

The Acme Standard or 29° thread is used for the same general purpose as 
the square thread. Threads per inch are likewise usually the same. In drawing, 
the angle is made 30°. 

In the Whitworth or British Standard, threads per inch are, with a few excep- 
tions, the same as U. S. St'd. See handbooks. 



126 



ESSENTIALS OF MECHANICAL DRAFTING 



(d) Conventional Representation of Threads. The true drawing of 
the thread curves involves considerable labor and in small screws would be 
impossible. In large threads it is customary to substitute st. lines for the helices, 
as in Fig. 217. In invisible threads they are often omitted altogether. 

V threads less than one inch diam. are usually represented as in Fig. 218 (a), 
(b), (c), (d). The methods (e), (f), (g), (h) are also used. The spacing is 
estimated by eye, without regard to the actual no. of threads per inch, but in 
methods (a), (b), (c), (d) the positions of the lines should indicate whether 
the screw is right- or left-hand. The thread of a long piece may be shown as 
in (i). 



V THREAD 




SQUARE THREAD 



ACME THREAD SINGLE R.H. V TH'D HOLE 



Double. 




Double 




Double 



Fig 217 




Sect ion 



Small square threads are usually represented as in (j). The exact no. 
of threads per inch is shown unless the scale is very small. 

A threaded hole is generally represented in the circular view as in (k), 
to distinguish it from a drilled hole; a drilled and threaded hole, as in (1). 

The diam. of the outer O in each is equal to that of the bolt or screw; that 
of the smaller about equal to that of the root. Any of the methods of Fig. 218 
may be used for the other views. On crowded drawings it is often best to use 
methods (d), (f), (h), or (m). It is better to omit the drawing of the threads 
beyond the screw end (see (n) ), unless method (m) is used. The point made 
by the drill is usually shown. 

Figs, (o) and (p) show methods of representing small pieces in sec. when 
V-threaded inside and outside. 

(e) Dimensioning. Give the outside diam.; the no. of threads per inch, 
thus: lOTh, lOThds., lOP., 10, or X: and the length of the threaded portion, 
from the end when chamfered, and from the curve when rounded. If the thread 
is other than right-hand and single, specify as indicated in Figs, (b), (c). 



HELICAL CUR\'ES, THREADED PARTS, AND SPRINGS 



127 




^ Pipe. Tap. 

Fig. 218 



4r" Pipe Tap. 



128 



ESSENTIALS OF MECHANICAL DRAFTING 



In a threaded hole, give the depth, the diam. of the piece to be screwed into 
it, and no. of threads per inch. Indicate diam. and no. of threads of a tapped 
hole, as in Figs, (k), (1). 

All parts shown V-threaded are generally understood to be U. S. St'd, unless 
otherwise specified; likewise when the no. of thds. is not given. 

111. Pipe Threads. The threaded ends and holes of pipes and pipe fittings 
are tapered so that the parts may be screwed together more tightly and thus 
prevent leakage. The standard taper is |" per foot. The thread Z is 60°, 
and the tops and bottoms are slightly rounded or flattened. The thread is 
usually represented by the conventional methods used for V screw threads. 
See Fig. 218 (q), (r). 

The taper is commonly drawn slightly greater than the actual, to show at 
a glance that the threads are pipe threads. The sizes of pipes are stated by 
giving their nominal inside diams., which are somewhat less than the actual 
inside diams., as noted in the table. Pipe tapped holes are indicated by size of 
pipe tap required. 

Standard Wrought Iron Pipe 



Nominal Inside 
Diam 



Actual Inside 
Diam 

Actual Outside 
Diam 



Threads per In. 

Diam. at Top of 
Thread at End 



1 

4 


f 


h 


3 
4 


1 


H 


1| 


2 


91 

'^2 


3 


^ 


4 


41 


5 


.36 


.49 


.62 


.82 


1.05 


1.38 


1.61 


2.07 


2.47 


3.07 


3.55 


4.03 


4.51 


5.05 


.54 


.68 


.84 


1.05 


1.32 


1.66 


1.90 


2.38 


2.88 


3.50 


4.00 


4.50 


5.00 


5.56 


18 


18 


14 


14 


111 


111 


111 


HI 


8 


8 


8 


8 


8 


8 


.52 


.62 


.82 


1.03 


1.28 


1.63 


1.87 


2.34 


2.82 


3.44 


3.94 


4.44 


4.93 


5.49 



6 
6.07 

6.63 

8 

6.55 



112. Bolts. The heads and nuts of machine bolts in common use are made 
hexagonal, or square, as in Fig. 219. 

The hexagonal form is more generally used in machine construction, the 
square in heavy work. For ordinary work, the head and nut are chamfered 
or beveled at the outer end. For finished machinery they are usually rounded. 

(a) Standard Proportions. Proportions of the U. S. St'd rough bolt- 
heads and nuts are given in the figure. They are generally used for the square 
also. There is no standard for the rad. R, nor for the bevel of the chamfers, 
but they are usually shown as in the figure. 

(b) Conventional Representations of Hexagon Heads and Nuts. In 
the rounded head and nut, the curves of intersection of the sides and end are 
circular. In the view across corners, therefore, the curve c-d is concentric with 
a-b, while those of the oblique sides would be elliptic. Art. 77(a). The latter, 
however, are always described as circular. 

Note that the outer curve in the nut begins at the hole instead of at the C. L. 



HELICAL CUR^-ES, THREADED PARTS, AND SPRINGS 



129 





R,= 2D 

R3=7-9 



Rounded 






[ 4! 12 6 .1 



D— 







. Jrt2Z 



HEXAGON HEAD BOLTS AND NUTS 




Rounded 



riiT 



/\ 



Proportions of U.S.SrD Rough Colts 
D= Nominal Diame+er of Bolt 
F = Width across Flats = l|D + 5" 
H = Thickness of Head=^F 
N= Thickness of Nut = D . 
Finished heads and nuts 
are i° less in width and thickness. 



/<o„>^ Chamfered 






square head bolts and nuvs bolt with check nut & washer. 

Fig. 219 



130 



ESSENTIALS OF MECHANICAL DRAFTING 



In the chamfered head and nut the curves of intersection of the sides and' 
end, though in reahty hyperbohc (Art. 77(c)), are likewise always described as 
circular. The outer Hne of the chamfer is often described by arcs concentric 
with a-b, as shown. 




HEXAGON HEAD BOLTS AND NUTS 



Fig 219(a) 



(c) To DRAW THE VIEW ACROSS CORNERS OF THE ROUNDED HEXAGON 

HEAD AND NUT. Upon an indefinite line 4-5 set off 1-2 equal to J F. Draw 
the _L 2-3 and the 30° hne from 1. Then 2-3 will represent half of a revolved 




TAP SCREW 




1 


— 




•a 


-p. 
Q 


1 


; 


1 








M 


-B- 


-h- 








t 











Sq. Head 

On Hex. and Square 
I T=|L When L=4"o(- less 
1 . =2L When Lis over 4" 



Hex.lHead 



CAP SCREWS 



Fig. 220 



HELICAL CURVES, THREADED PARTS, AND SPRINGS 



131 



side of the hexagon and 1-3 half of its long diain. Now set off 1-4 and 1-5 equal 
to 1-3, and 1-6 and 1-7 equal to 2-3, and draw 4-a, 5-b, 6-c, and 7-d. Next 
set off H and draw arc a-b determining the length of the ±s. Finally, draw 
arcs c-d, a-c, and d-b. In small drawings the long diam. may be made equal 
to 2 D. The method of drawing the nut is evident. 

(d) To DRAW THE VIEW ACROSS FLATS OF THE ROUNDED HEXAGON HEAD 

AND NUT. Set off 7-8 equal to F, and draw 7-e and 8-f. Next set off H and 

draw arc e-f determining pts. e and f. Then determine pts. d, b, and g, as in 

Art. (c), and describe arcs d-b and b-g.. The method of drawing the nut is 

evident. 

machine: screws 
Flat Head Round Head 





\X7J CO 



Cone Pt. Flat Pt. Cup Pt. Round Pt. 

se.t screws 
Fig. 221 



Pivot Pt. 



Fillistai- Head 
Fiat Oval 






■*— ^ i ' ^ 



Fig. 222 



(e) To DRAW THE CHAMFERED HEAD AND NUT ACROSS 

CORNERS OR ACROSS FLATS. The method for each is 
evident from Arts, (c) and (d). 

(f) Square Heads and Nuts. The method of 

drawing the square head and nut is, in general, the same as for the hexagon. 

(g) When drawn in connection with the parts held together, both heads and 
nuts should, as a rule, be represented across corners to show that proper allow- 
ance has been made for clearance; otherwise they should be shown aci'oss flats, 
as they are thus simpler to draw and to figure. 

(h) Fig. 219 also illustrates a st'd hexagon bolt with check nut and washer. 
Both nuts are often made equal in thickness, f D. 

(i) Dimensioning. In a st'd bolt give the diam.; length of bolt from the 
under side of the head to extreme end, unless the end is rounded; and the length 
of the threaded portion. 

In a special bolt give also the distance across flats, the thickness of head and nut 
and the no. of thds. per inch. 

113. Screws. Fig. 220 represents a tap screw or tap holt; a stud holt or stud, 
and hexagon and square head cap screivs. 

A tap screw is similar to a st'd bolt without the nut; the end being screwed 
into a tapped hole. 



132 



ESSENTIALS OF MECHANICAL DRAFTING 



A stud is used where frequent removal is not desirable, as in cylinder heads. 
One end is screwed permanently into a tapped hole and a st'd nut used on the 
other. A cap screw is a type of tap screw used where adjustment is necessary, 
as on bearing caps, etc. 

Fig. 221 represents set screws which are used to prevent the motion of one 
piece by forcing the point against a second. 

The form of point used is dependent upon the resistance desired. 

Fig. 222 represents four types of machine screws which are from .06" to 
.45" in diam. and designated by gage number. Slots are drawn at 45° in end 
views for contrast with other lines. Tables of proportions of these and other 
forms of screws and bolts may be found in catalogs and engineers' hand- 
books. Fig. 223 represents wood screws. 





'^^^^^r^ 



Flat H'd R'd M'd 

\X7 CO 





wood scre.ws 
Fig 223 



helical springs 
Fig. 224 



114. Springs. Fig. 224 shows conventional representations of /ieh'caZ spnngfs. 
In small sees, the helical lines are often omitted. 

In dimensioning, give outside diam., gage of wire, and coils per inch when 
extended. 



